a core course on modeling

A Core Course on Modeling

The modeling process

define

conceptualize

conclude

execute

formalize

formulate

purpose

formulate

purpose

identify

entities

identify

entities

choose

relations

choose

relations

obtain

values

obtain

values

formalize

relations

formalize

relations

operate

obtain

result

obtain

result

present

result

present

result

interpret

result

interpret

result

Week 1- No Model Without a Purpose

Right problem?

Right concepts?

Right model?

Right outcome?

Right answer?Right answer?Right answer?Right answer?

Contents • What is a Formal Model?

• A Practical Route to Formal Models

• Example 1: The Detergent Problem

• Example 2: The Chimney Sweepers Problem

• Example 3: The Peanut Butter Problem

•The relation wizard

•The function selector• Summary

• References to lecture notes + book

• References to quiz-questions and homework assignments (lecture notes)

Week 4-The Function of Functions

What is a Formal Model? What is the meaning of + ?

resistors: Rtot = R1 + R2

springs: Ctot = 1/(1/C1+1/C2)

resistors: Rtot = 1/(1/R1 + 1/R2)

springs: Ctot = C1 + C2

the intuition of ‘addition’, ‘accumulation’:

A Practical Route to Formal Models Heuristics to arrive at formal expressions:

• meaningful names

• chain of dependencies

• todo-list

• dimensional analysis

• wisdom of the crowds

The Detergent Problem

“What is the total amount of detergent

annually dumped in the

Environment in the Netherlnds?

amAnDetDmp =

f(nrAnWshs , detPWsh)

relations dimensions assumptions todo[kg / year] =

F([wash / year] , [kg / wash])

amAnDetDmp

nrAnWshs

detPWsh

Det:detergent; An: annual; Wsh: wash; Fam: family; P: people; am: amount; Dmp: dump

amAnDetDmp =

nrAnWshs * detPWsh

[wash / year] * [kg / wash]amAnDetDmp

nrAnWshs

detPWsh

amAnDetDmp =

nrAnWshs * detPWsh

[wash / year] * [kg / wash]

Washing laundry is the only

way detergent gets into the

environment

amAnDetDmp

nrAnWshs

detPWsh

nrAnWshsPFam

nrFamIH

nrPPFam

nrAnWshs =

nrAnWshsPFam * nrFamIH

[wash / year] =

[wash / (fam *year)] * [fam]

No institutional laundry washing, only families

nrFamIH =

nrPIH / nrPPFam

[fam] =

[people] / [people / fam]

Everybody belongs to exactly one family: families are disjoint

nrPIH = 17 0.5 million [people] common knowledge

nrPPFam = 2.2 0.2 [people/fam] public domain

nrAnWshsPFam = ????? [wash / year] wisdom of the crowds

detPWsh = 0.17 0.03 [kg / wash] wisdom of the crowds

todo list is empty model is ready

amAnDetDmp

nrAnWshs

detPWsh

nrAnWshsPFam

nrFamIH

nrPPFam

amAnDetDmp

nrPPFamnrPIH

nrFamIH

nrAnWshsPFam

detPWshnrAnWshs *

amAnDetDmp

nrAnWshs

detPWsh

nrAnWshsPFam

nrFamIH

nrPPFam

amAnDetDmp

nrPPFamnrPIH

nrFamIH

nrAnWshsPFam

detPWshnrAnWshs *

170.5 million2.20.2

7.72 … million

0.170.03

1.31… million

772…million

The Chimney Sweepers Problem

“How many chimney sweepers

work in Amsterdam?”

nrChSwIA =

nrChIA * nrSwPCh

relations dimensions assumptions todo

[Sw / A] =

[Ch / A] * [SW / Ch]

Amsterdam ch.-sweepers

sweep Amsterdam

chimneys only

nrChSwIA

nrChIA

nrSwPCh

nrChPFam

nrFamIA

nrPPFam

nrChIA =

nrChPFam * nrFamIA

[Ch / A] =

[Ch / Fam] * [Fam / A]

ch.-sweepers sweep only chimneys on family houses

nrFamIA =

nrPIA / nrPPFam

[Fam / A] =

[P / A] / [P / Fam]

nrPPFam is the same everywhere (does not depend on ‘Amsterdam’)

nrPIA = 790000 [P] common knowledge

nrPPFam = 2.2 0.2 [P/Fam] public domain

nrChPFam (= 1/nrFamPCh) =0.10.02 [Ch/Fam]

wisdom of the crowds

Sw=sweeper; Ch=chimney;A=Amsterdam;Fam=Family;P=people;Se=Service;

nrSwPCh =

nrSwPSe * nrSePCh

[Sw / Ch] =

[Sw*year/Se] * [Se/(Ch*year)]

Introduce time to associate sweeper’s ca-pacity to chimney’s need

nrChSwIA

nrChIA

nrSwPCh

nrChPFam

nrFamIA

nrPPFam

nrSwPSe

nrSePCh

timeP1Se

timeP1Sw

nrSwPSe =

timeP1Se / timeP1Sw

[Sw * year / Se] =

[hour / Se] / [hour / (Sw*year)]

assume average times (i.e., no season influences

timeP1Sw = 1200100 hour / Sw * year)

Sw=sweeper; Ch=chimney;A=Amsterdam;Fam=Family;P=people;Se=Service;

timeP1Se = 20.25 hour / Se wisdom of the crowds

work year = 1600 hours

nrSePCh = 1 Se /( Ch * year) insurance requirement

todo list is empty model is ready

but what does it mean ? NOTHING, since we formulated no purpose

What purposes could we think of:

begin a professional journal …

so: we only need to know if NrChSwIA > 300

ChSw convention meeting in the Restaurant ‘the Swinging Sweeper?’

so: we only need to know if NrChSwIA <50

form efficient ‘Chimney and Sewage Control and Service Units’?

so: we only need to know if NrChSwIA is between 50 and 60…

To assess credibility of a model:

•actual measurements ?

•second, independent model ? Such as …

• how much soot and ashes are disposed of by the municipal Ash & Soot Depot?

• how often do you see a chimney sweeper at work?

• how much money do people in A. spend annually in cleaning their chimneys?

check latter case:

The Peanut Butter Problem

“How to get rich

by selling a new brand

peanut butter?”

income =

pricePerItem * nrSoldItems

[Euro / year] =

[Euro/myPB] * [myPB/year]

no discount with larger quantities per purchase

profit

income

expenses

pricePerItem

nrSoldItems

nrSoldTotal

marketSharenrSoldItems =

nrSoldTotal * marketShare

[myPB / year] =

[allPB / year] * [myPB/allPB]

my PB will not increase the total market

pricePerItem = …

nrSoldTotal = … [allPB/year] from a neutral marketing bureau

marketShare = …

profit = income - expenses [Euro / year] = [Euro/year] no taxes, no inflation

Approach 1: glass box

pricePerItem

What mechanism determines marketShare(pricePerItem)?• monotonically decreasing• between 0 and 1• asymptote: marketShare0 if pricePerItem • asymptote: marketShare1 if pricePerItem - • what sort of mathematical dependency ???

pricePerItem

cheapestcompetitor

most expensivecompetitor

pricePerItem

cheapestcompetitor

pricePerItem

cheapestcompetitor

pricePerItem

Approach 2: black box

pricePerItem

Approach 2: black box

Conceptual model formal model:

while (purpose not satisfied):

•identify quantity q needed for purpose•q to-do list•while todo (not empty):

•take a quantity r from todo list•think: what does r depend on?

•if r independent substitute constant value with uncertainty bounds

•else give expression for r•if possible, use dimensional analysis•propose mathematical expression r = f(s1, s2, s3, …)

•think about assumptions•verify dimensions

•add newly introduced quantities s1, s2, s3, … to the todo list

•todo list is empty: evaluate model•check if purpose is satisfied; if not, refine model

• ready

To get help in the translation to mathematical relations:

relation wizard

function selector

pricePerItem

cheapestcompetitor

• Conceptual model formal model : not in a formally provable correct way;

• Appropriate naming

• Structure

• Chain of dependencies: the formal model as a directed acyclic graph;

•What mechanism?

•What quantities drive this mechanism?

•What is the qualitative behavior of the mechanism?

•What is the mathematical expression to describe this mechanism?

• To-do-list : all intermediate quantities are found and elaborated in turn;

• Formation of mathematical expressions:

•dimensional analysis mathematical expressions, e.g in the case of proportionality

•the Relation Wizard can help finding appropriate fragments of mathematics;

•the Function Selector can help finding an appropriate expression for a desired behavior;

•wisdom of the crowds can help improve the accuracy of guessed values;

Summary

Peanut Butter example:

income = pricePerItem * nrSoldItems

nrSoldItems = nrSoldTotal * marketShare

marketShare = f(pricePerItem)

Revisit the peanut butter example:

inc = pPI * nSI (inc=income; pPI=pricePerItem; nSI=nrSoldItems)

nSI = nST * mSh (nST=nrSoldTotal; mSh=marketShare)

mSh = f(pPI)

So: inc = pPI * nST * mSh(pPI)

Realized that nST = i nST(i), i ranges over customers.

So: inc = pPI * mSh(pPI) * i nST(i)

Improved model: inc = pPI * i nST(i) * mSh(pPI , i)

a core course on modeling

detergent problem week

detergent problemexample

way detergent

wash fam

nrfamihwash year

fwash year

modeling contents

people people fameverybody

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