Creative Modeling for Technology Visionaries Qualitative & Simplified Quantitative Modeling for Product Creation Module 10: Crossing the Qual-Quant Divide David E. Goldberg University of Illinois at Urbana-Champaign Urbana, Illinois 61801 [email_address]

From Qual to Quant Started with qualitative models, verbal & visual. Want to go quantitative for a variety of reasons. Why do we do so? How do we do so, especially in situations where equations of motion do not exist? Needs sense of number, ratio, and dimension.

Started with qualitative models, verbal & visual.

Want to go quantitative for a variety of reasons.

Why do we do so?

How do we do so, especially in situations where equations of motion do not exist?

Needs sense of number, ratio, and dimension.

This Module Recall modeling economics and spectrum. Why go quantitative? A short history of numbers. Measuring things: ratios, ratios, everywhere. Graphical representations of measurements. The link to qualitative modeling: dimensions. A short history of algebra. A menagerie of models: deterministic and probabilistic. Different approaches to combining models.

Recall modeling economics and spectrum.

Why go quantitative?

A short history of numbers.

Measuring things: ratios, ratios, everywhere.

Graphical representations of measurements.

The link to qualitative modeling: dimensions.

A short history of algebra.

A menagerie of models: deterministic and probabilistic.

Different approaches to combining models.

Fundamental Modeling Tradeoff Error versus cost of modeling ε , Error C, Cost of Modeling Engineer/Inventor Scientist/Mathematician

Error versus cost of modeling

Spectrum of Models

Why Go Quantitative? For many product applications, developing pictures and language to describe products is crucial. Organized dimensionalization of product space a key. Competition can drive quantitative development: More precise prediction in design. Reduce uncertainty of design outcomes. Better qualitative understanding.

For many product applications, developing pictures and language to describe products is crucial.

Organized dimensionalization of product space a key.

Competition can drive quantitative development:

More precise prediction in design.

Reduce uncertainty of design outcomes.

Better qualitative understanding.

Precision & Uncertainty Going quantitative improves precision. Can reduce uncertainty of design outcomes. Well understood as key driver of engineering knowledge. Views accretion of engineering knowledge as evolutionary process to reduce uncertainty.

Going quantitative improves precision.

Can reduce uncertainty of design outcomes.

Well understood as key driver of engineering knowledge.

Views accretion of engineering knowledge as evolutionary process to reduce uncertainty.

Going Quant for Qual More detailed quantitative modeling can improve qualitative understanding. Variations of key quantities can be used to explain phenomena. Example: Stall & flaps at a glance.

More detailed quantitative modeling can improve qualitative understanding.

Variations of key quantities can be used to explain phenomena.

Example: Stall & flaps at a glance.

A Short History of Numbers 57 notches on wolf bone 30k years ago in groups of five. Ishango bone from Zaire 6-9kya (at right). Egyptian (3k BC) & Mesopotamian math (2k BC).

57 notches on wolf bone 30k years ago in groups of five.

Ishango bone from Zaire 6-9kya (at right).

Egyptian (3k BC) & Mesopotamian math (2k BC).

Measuring Stuff Ancient length units used body parts as reference: the digit (width of the forefinger - in Latin digitus = finger) the inch (width of the thumb) the foot the cubit ( theoretically the distance between the elbow and the middle finger) the pace (or double step) the fathom (finger-tip to finger-tip with arms outstreched) Key idea is notion of ratio to some standard.

Ancient length units used body parts as reference:

the digit (width of the forefinger - in Latin digitus = finger)

the inch (width of the thumb)

the foot

the cubit ( theoretically the distance between the elbow and the middle finger)

the pace (or double step)

the fathom (finger-tip to finger-tip with arms outstreched)

Key idea is notion of ratio to some standard.

Length, Time, & All That From length, to time, to units of physics. We measure these things in units. Each one is a physical dimension. Can break quantities down into component dimensions. Borrow from dimensional analysis, bracket notation. [V] = L/T (velocity has dimensions of length per time).

From length, to time, to units of physics.

We measure these things in units.

Each one is a physical dimension.

Can break quantities down into component dimensions.

Borrow from dimensional analysis, bracket notation.

[V] = L/T (velocity has dimensions of length per time).

Functions & Algebra Used since ancient times. Tables of functions used by Egyptians and Babylonians. Egyptians could solve linear equations. Early Hindu algebra almost symbolic. Symbolic algebra only in 1500s. Term “function” coined by Leibniz in 1694.

Used since ancient times. Tables of functions used by Egyptians and Babylonians.

Egyptians could solve linear equations.

Early Hindu algebra almost symbolic.

Symbolic algebra only in 1500s.

Term “function” coined by Leibniz in 1694.

Qual-Quant: Dimensions & Functions How do we move from qual to quant? Take qualitative dimensions and assume existence of a function. Construct empirically or theoretically. This as a function of that y=f(x). This as a function of that, that, & that w=f(x,y,z).

How do we move from qual to quant?

Take qualitative dimensions and assume existence of a function.

Construct empirically or theoretically.

This as a function of that y=f(x).

This as a function of that, that, & that w=f(x,y,z).

Dimensional Analysis Engineers use dimensional analysis to understand technical objects: e.g., lift coefficients & Reynold’s numbers for flow analysis. Based on idea of dimensionless ratio: Lift Coefficient = ratio of lift to inertial force. Dynamic similitude: Dynamically similar objects should have identical dimensionless ratios. Lift Coefficients

Engineers use dimensional analysis to understand technical objects: e.g., lift coefficients & Reynold’s numbers for flow analysis.

Based on idea of dimensionless ratio: Lift Coefficient = ratio of lift to inertial force.

Dynamic similitude: Dynamically similar objects should have identical dimensionless ratios.

Empirical Functions Plot function from data. Possibly consider a curve fit. Advantage: Matches in situ system. Disadvantage: Can be expensive to collect data .

Plot function from data.

Possibly consider a curve fit.

Advantage: Matches in situ system.

Disadvantage: Can be expensive to collect data .

Theoretical Forms Simple deterministic forms: Constant Linear Power law Exponential or logarithmic Ways of integrating: Summing Patching

Simple deterministic forms:

Constant

Linear

Power law

Exponential or logarithmic

Ways of integrating:

Summing

Patching

The Power of Simple Forms A constant can be quite powerful, (especially with dimensional analysis). Linear relations used all the time. Constant models: Financial ratios. Linear models: breakeven analysis.

A constant can be quite powerful, (especially with dimensional analysis).

Linear relations used all the time.

Constant models: Financial ratios.

Linear models: breakeven analysis.

Financial Ratio Analysis Entrepreneurial engineers use ratio analysis to understand financial objects (businesses). Dimensional analysis for business. Market similitude: Market-dynamically similar businesses should have similar financial ratios. Categorize “typical” ratios by market, size, other characteristics.

Entrepreneurial engineers use ratio analysis to understand financial objects (businesses).

Dimensional analysis for business.

Market similitude: Market-dynamically similar businesses should have similar financial ratios.

Categorize “typical” ratios by market, size, other characteristics.

Price-to-Earnings Ratio P/E P/E = Current market price / Earnings per share. Way to judge whether stock is good value relative to others. Reciprocal of P/E may be thought of as an interest rate: Suppose all income paid out as a dividend. Most companies retain earnings.

P/E = Current market price / Earnings per share.

Way to judge whether stock is good value relative to others.

Reciprocal of P/E may be thought of as an interest rate:

Suppose all income paid out as a dividend.

Most companies retain earnings.

Other Ratios Return on sales (ROS): Net income / sales revenue. From pennies in groceries to significant fraction. Return on equity (ROE): Net income / shareholder’s equity. Is return greater than less risky investment? Debt-to-equity ratio: Total liabilities / shareholder’s equity. Less than one, lenders have more to lose than owners.

Return on sales (ROS):

Net income / sales revenue.

From pennies in groceries to significant fraction.

Return on equity (ROE):

Net income / shareholder’s equity.

Is return greater than less risky investment?

Debt-to-equity ratio:

Total liabilities / shareholder’s equity.

Less than one, lenders have more to lose than owners.

Simple Breakeven Analysis Assume P, price per unit F, fixed cost V, variable cost n, number of units sold Pr, profit Revenue calculation straightforward Pr = nM – F where M = P – V.

Assume

P, price per unit

F, fixed cost

V, variable cost

n, number of units sold

Pr, profit

Revenue calculation straightforward Pr = nM – F where M = P – V.

Breakeven n b = F/M, where n b is the breakeven volume. Pr = ( n – n b ) M Key is to think in terms of contribution margin.

n b = F/M, where n b is the breakeven volume.

Pr = ( n – n b ) M

Key is to think in terms of contribution margin.

Visualizing Breakeven 0 Pr n b - F n Slope M = P - V

Probabilistic Models Independent events: p T = p 1 p 2 p 3. Time to expected occurrence of all 3 things: T T = 1/p T . Success of n independent trials of Bernoulli process with success probability p: P = 1 – (1 – p ) n .

Independent events: p T = p 1 p 2 p 3.

Time to expected occurrence of all 3 things: T T = 1/p T .

Success of n independent trials of Bernoulli process with success probability p: P = 1 – (1 – p ) n .

A Model of Innovation Borrowed from my genetic algorithm work, but applicable more generally. Consider population of consumers choosing better product with pressure s. Consider market creating innovations at certain rate.

Borrowed from my genetic algorithm work, but applicable more generally.

Consider population of consumers choosing better product with pressure s.

Consider market creating innovations at certain rate.

Time to Market Takeover of Best Best has pressure s. Get s copies of best in population n. Called truncation selection. Truncation selection: make s copies each of top 1/ s th of the population. Want to know time for good individual to “takeover” population: takeover time.

Best has pressure s.

Get s copies of best in population n.

Called truncation selection.

Truncation selection: make s copies each of top 1/ s th of the population.

Want to know time for good individual to “takeover” population: takeover time.

Takeover Time Under Truncation Let P be proportion of best individuals. How many more next generation? s P ( t +1) = s P ( t ) until P ( t )=1. P ( t ) = s t P (0). Solve for takeover time t *: time to go from one good guy to all or all but one good guys. t* = ln n/ ln s. Sometimes this process of isolating a facet is called facetwise modeling.

Let P be proportion of best individuals.

How many more next generation? s

P ( t +1) = s P ( t ) until P ( t )=1.

P ( t ) = s t P (0).

Solve for takeover time t *: time to go from one good guy to all or all but one good guys.

t* = ln n/ ln s.

Sometimes this process of isolating a facet is called facetwise modeling.

Consider Innovation of Firms Assume firms attempt to innovate in generation with probability p c . Assume that innovation trial gives improvement with probability p i in population of size n. Can calculate the expected time of arrival of next improvement.

Assume firms attempt to innovate in generation with probability p c .

Assume that innovation trial gives improvement with probability p i in population of size n.

Can calculate the expected time of arrival of next improvement.

The Innovation Time, t i Define innovation time as the expected time to create an individual better than one so far. t i = ( p c p i n ) -1 . Model is facetwise, probabilistic & incomplete ( p i unknown). p i estimated in Goldberg, Deb, & Thierens (1993) and Thierens & Goldberg (1993).

Define innovation time as the expected time to create an individual better than one so far.

t i = ( p c p i n ) -1 .

Model is facetwise, probabilistic & incomplete ( p i unknown).

p i estimated in Goldberg, Deb, & Thierens (1993) and Thierens & Goldberg (1993).

The Race: Integrating 2 Models Have two facetwise models, but want integrated understanding. Putting models in t terms gives us an idea. Consider relative magnitudes of the two times: a dimensional argument. Consider which is favorable to innovation: a qualitative argument.

Have two facetwise models, but want integrated understanding.

Putting models in t terms gives us an idea.

Consider relative magnitudes of the two times: a dimensional argument.

Consider which is favorable to innovation: a qualitative argument.

Schematic of the Race

Consider Ratio of Two Times Define innovation number G = t* / t i G = p c p i n ln n/ ln s. Want takeover time greater than innovation time or G > 1. Quantity like Reynolds number in fluids. Argument was a dimensional argument.

Define innovation number G = t* / t i

G = p c p i n ln n/ ln s.

Want takeover time greater than innovation time or G > 1.

Quantity like Reynolds number in fluids.

Argument was a dimensional argument.

A Simple Control Map Draw success region in GA parameter space. Control map. p c > c ( m, n ) ln s. For p c versus ln s a straight line .

Draw success region in GA parameter space.

Control map.

p c > c ( m, n ) ln s.

For p c versus ln s a straight line .

Sweet Spot for Easy Problem Easy problems are no problem. Innovation can have a large “sweet spot”. Notice “patching” of other models.

Easy problems are no problem.

Innovation can have a large “sweet spot”.

Notice “patching” of other models.

Bottom Line Socratic questions how and why cross the qual-quant divide. Considered importance of number, units, dimensions, and functions. Various ways of constructing functional relationships from constants on up. Deterministic and probabilistic. Modeling easy to take for granted, easy to make fancier than necessary.

Socratic questions how and why cross the qual-quant divide.

Considered importance of number, units, dimensions, and functions.

Various ways of constructing functional relationships from constants on up.

Deterministic and probabilistic.

Modeling easy to take for granted, easy to make fancier than necessary.