Download - A Core Course on Modeling
1
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
model
operate
model
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?
2
A Core Course on Modeling
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
3
A Core Course on Modeling
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’:
Week 4-The Function of Functions
4
A Core Course on Modeling
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
Week 4-The Function of Functions
5
A Core Course on Modeling
The Detergent Problem
“What is the total amount of detergent
annually dumped in the
Environment in the Netherlnds?
Week 4-The Function of Functions
6
A Core Course on Modeling
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
The Detergent Problem
Week 4-The Function of Functions
7
A Core Course on Modeling
amAnDetDmp =
nrAnWshs * detPWsh
relations dimensions assumptions todo[kg / year] =
[wash / year] * [kg / wash]amAnDetDmp
nrAnWshs
detPWsh
Det:detergent; An: annual; Wsh: wash; Fam: family; P: people; am: amount; Dmp: dump
The Detergent Problem
Week 4-The Function of Functions
8
A Core Course on Modeling
amAnDetDmp =
nrAnWshs * detPWsh
relations dimensions assumptions todo[kg / year] =
[wash / year] * [kg / wash]
Washing laundry is the only
way detergent gets into the
environment
amAnDetDmp
nrAnWshs
detPWsh
nrAnWshsPFam
nrFamIH
nrPIH
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
Det:detergent; An: annual; Wsh: wash; Fam: family; P: people; am: amount; Dmp: dump
detPWsh = 0.17 0.03 [kg / wash] wisdom of the crowds
todo list is empty model is ready
The Detergent Problem
Week 4-The Function of Functions
9
A Core Course on Modeling
todo
amAnDetDmp
nrAnWshs
detPWsh
nrAnWshsPFam
nrFamIH
nrPIH
nrPPFam
The Detergent Problem
Week 4-The Function of Functions
amAnDetDmp
nrPPFamnrPIH
nrFamIH
nrAnWshsPFam
detPWshnrAnWshs *
/
*
10
A Core Course on Modeling
todo
amAnDetDmp
nrAnWshs
detPWsh
nrAnWshsPFam
nrFamIH
nrPIH
nrPPFam
The Detergent Problem
Week 4-The Function of Functions
amAnDetDmp
nrPPFamnrPIH
nrFamIH
nrAnWshsPFam
detPWshnrAnWshs *
/
*
170.5 million2.20.2
7.72 … million
0.170.03
1.31… million
772…million
10020
11
A Core Course on Modeling
The Chimney Sweepers Problem
“How many chimney sweepers
work in Amsterdam?”
Week 4-The Function of Functions
12
A Core Course on Modeling
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
nrPIA
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;
The Chimney Sweepers Problem
Week 4-The Function of Functions
13
A Core Course on Modeling
nrSwPCh =
nrSwPSe * nrSePCh
relations dimensions assumptions todo
[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
nrPIA
nrPPFam
nrSwPSe
nrSePCh
timeP1Se
timeP1Sw
nrSwPSe =
timeP1Se / timeP1Sw
[Sw * year / Se] =
[hour / Se] / [hour / (Sw*year)]
assume average times (i.e., no season influences
etc.)
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
The Chimney Sweepers Problem
Week 4-The Function of Functions
14
A Core Course on Modeling
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…
The Chimney Sweepers Problem
Week 4-The Function of Functions
15
A Core Course on Modeling
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 Chimney Sweepers Problem
Week 4-The Function of Functions
16
A Core Course on Modeling
The Peanut Butter Problem
“How to get rich
by selling a new brand
of
peanut butter?”
PB
Week 4-The Function of Functions
17
A Core Course on Modeling
income =
pricePerItem * nrSoldItems
relations dimensions assumptions todo
[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 = …
The Peanut Butter Problem
Week 4-The Function of Functions
profit = income - expenses [Euro / year] = [Euro/year] no taxes, no inflation
18
A Core Course on Modeling
Approach 1: glass box
pricePerItem
mar
ketS
hare
0
1
??
The Peanut Butter Problem
Week 4-The Function of Functions
19
A Core Course on Modeling
Approach 1: glass box
pricePerItem
mar
ketS
hare
0
1
What mechanism determines marketShare(pricePerItem)?• monotonically decreasing• between 0 and 1• asymptote: marketShare0 if pricePerItem • asymptote: marketShare1 if pricePerItem - • what sort of mathematical dependency ???
The Peanut Butter Problem
Week 4-The Function of Functions
20
A Core Course on Modeling
Approach 1: glass box
pricePerItem
mar
ketS
hare
0
1
cheapestcompetitor
most expensivecompetitor
The Peanut Butter Problem
Week 4-The Function of Functions
21
A Core Course on Modeling
Approach 1: glass box
pricePerItem
mar
ketS
hare
0
1
cheapestcompetitor
most expensivecompetitor
The Peanut Butter Problem
Week 4-The Function of Functions
22
A Core Course on Modeling
Approach 1: glass box
pricePerItem
mar
ketS
hare
0
1
cheapestcompetitor
most expensivecompetitor
The Peanut Butter Problem
Week 4-The Function of Functions
23
A Core Course on Modeling
pricePerItem
mar
ketS
hare
0
1
The Peanut Butter Problem
Week 4-The Function of Functions
Approach 2: black box
24
A Core Course on Modeling
pricePerItem
mar
ketS
hare
0
1
The Peanut Butter Problem
Week 4-The Function of Functions
Approach 2: black box
25
A Core Course on Modeling
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
The Peanut Butter Problem
Week 4-The Function of Functions
26
A Core Course on Modeling
The Peanut Butter Problem
Week 4-The Function of Functions
To get help in the translation to mathematical relations:
relation wizard
function selector
27
A Core Course on Modeling
Approach 1: glass box
pricePerItem
mar
ketS
hare
0
1
cheapestcompetitor
most expensivecompetitor
The Peanut Butter Problem
Week 4-The Function of Functions
28
A Core Course on Modeling
• 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
Week 4-The Function of Functions
29
A Core Course on Modeling
Peanut Butter example:
income = pricePerItem * nrSoldItems
nrSoldItems = nrSoldTotal * marketShare
marketShare = f(pricePerItem)
The Peanut Butter Problem
Week 4-The Function of Functions
30
A Core Course on Modeling
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)
The Peanut Butter Problem
Week 4-The Function of Functions