Creative Modeling for
Technology Visionaries
Qualitative & Simplified Quantitative Modeling for Product Creation
Module 14: Mixed, Patched & Meta-Models
David E. Goldberg
University of Illinois at Urbana-Champaign
Urbana, Illinois 61801
deg@uiuc.edu

Making Little Modeling Fully Usable
Can sometimes squeeze models with
mixed terms.
Usually need to combine or patch
multiple models.
Can move beyond models of
phenomena and create meta-models.
Will have the basic tools for any
LMing task.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 2

This Module
2 mixed models:
Search or transaction costs.
Hierarchical organization.
Integrating models with patchquilts: Can we
integrate two or models?
Models of models:
Can we go quantitative about modeling itself?
Consider various tradeoffs of models themselves.
Short wrapup of course.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 3

The Problem of Mixed Models
Many of our models with explicit solution
have a particular form:
Some mixed cases, f and g unrelated.
2 examples: search & hierarchy.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 4

Searching for Things of Value at Cost
Searching for something of value.
Each trial has a cost C.
Each trial is successful with probability p.
When found object realizes value V.
Could be a job (transaction costs), an
employee, or other thing of value.
Optimize number of trials by maximizing
expected profit.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 5

Optimizing Transactions
Imagine
Transaction costs C
Success probability p
Marginal value of successful transaction V
Profit P of nth transaction:
P = V[1 – (1 – p )n] – Cn
Maximizing profit yields:
n* = [ln V/C + ln r] / r,
where q = 1 – p, r = -ln q
Example: p = 0.1, V = 10, C = 0.1, n* = 22.4.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 6

Trials vs. Search Success
10
profit
value
cost
8
Profit, Value, or Cost of Hire
6
4
2
0
-2
0 10 20 30 40 50 60 70 80 90 100
Annual Frequency of Hires
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 7

Turning Point Solution
Set Costs = Value.
V[1 – (1 – p )n] = Cn.
Explicit differential
solution easier. 10
profit
value
TP not bad 8
cost
approximation,
Profit, Value, or Cost of Hire
6
however. 4
2
0
-2
0 10 20 30 40 50 60 70 80 90 100
Annual Frequency of Hires
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 8

Imagine a Hierarchy
Bosses communicate down the hierarchy
to the bosses at the next level
sequentially.
Within teams there is communication with
specified topology.
What team size k minimizes the total
communication m.
T = T 1 k + T2 d
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 9

Optimal Span of Control
o m = total organization size.
o k = team size.
o d = organization depth.
o Minimize total coordination
communication in organization.
o
o
o
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 10

Results on Texas School Data
Does this work in real
organizations?
678 Texas schools from
1994 to 1997, total 2712
data points (m,k) [Moore &
Bohte, 2000, 2003].
Measuring the quality of
models:
k(lnk)2
Compute the average T1/T2.
Given the organization size
m, compute the SOC k’.
Compute the mean error |k-
k’|
m
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 11

Can Generalize Topology
Mean error
Can generalize to
arbitrary exponents.
Both linear and
quadratic are nearly
a
as predictive.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 12

A Simple Approach to Complexity
Complex systems complex, because they
have multiple facets.
Cannot expect simple model to model
then all.
Different tradeoffs govern in different
parts of “phase space.”
Can we build multiple models and
integrate them some how?
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 13

How to Integrate LMs
Integration methods:
Asymptotic methods of applied math &
physics.
Probabilistic blending.
Patchquilt integration.
First two add complexity of their own.
Let’s KISS (keep it simple stupid).
Consider patchquilt integration.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 14

Example from Fluids: Pipeflow
Laminar flows: model
is analytical.
Turbulent flow model
is partially analytical,
partially a curve fit.
Patchquilt is
discontinuous (region
of instability) yet
principled.
Moody diagram a patchquilt
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 15

Example from GAs: Sweet Spot
Mixing, drift, and cross-competitive
boundaries intersect.
Experiments match shape.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 16

Types of Patch
Transitions occur at critical values of a
dimensionless parameter.
Usually determined experimentally.
Rapid qualitative change in system.
Intersecting inequalities:
Boundaries of region determine feasibility.
Example: Sweet Spot.
Intersecting equalities:
Requires conflict management scheme.
Often bigger or smaller is better.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 17

A Patchquilt Approach
Organization
size: m
Model1:
Model2:
Larger value
governs (bigger
better conflict
resolution).
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 18

Meta-Models: Models of Models
Can we characterize models
mathematically?
Would like models that are optimum in
some sense.
Will consider two cases:
Theoretical modeling: Error vs. cost of
interpretation.
Empirical modeling: Reliability vs. cost of
sampling.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 19

Interpretation versus Ignorance
Little models are easily interpreted, but
sometimes they are not as accurate as
others.
This is the fundamental tradeoff of
modeling.
Considered in information theory as
minimum description length.
Let’s do bounding little models of little
models.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 20

Interpretation versus Ignorance #1
Suppose ignorance falls off inversely with
length: Cε/ℓ.
Cε
C= + C 

Suppose cost of interpretation goes up as
the number of terms: Cℓℓ.
Form is an old friend.
Cε
* =
C
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 21

Interpretation versus Ignorance #2
Suppose ignorance falls off geometrically with
model length:
ε = (1 − α )

Suppose cost of interpretation goes up as the
number of terms.
Form is essential same as search model.
C = Cε (1 − α ) + C 

Solution: 1 C where
* = − ln r = − ln 1 − α
r rCε
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 22

Consider Typical Lab Setting
Can sample more and improve accuracy
of mean measurement.
But each sample has a cost.
Stat classes teach more samples better,
but what if we can quantify cost of error
and cost of sample.
Wouldn’t we want to minimize total costs?
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 23

Sample Cost vs. Lowered Uncertainty
Error goes down inversely as σ
square root of number of samples, ε = 1/ 2
m. m
Assume cost increase is
linear.
Total cost: Cε σ
C = Cs m + 1/ 2
m
Optimum: 2/3
 Cε σ 
m* = 
 2C 

 s 
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 24

Bottom Line
Can sometimes handle mixed little
models.
Oftentimes can integrate LMs in
patches.
LMs of LMs give important insights.
What about larger lessons of course
about modeling generally.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 25

The Bottom, Bottom Line
Science is relentless in search of better knowledge:
Models are end.
Engineering is relentless in search of better designs:
Models are instrumental to that end.
Need for category creators shifts emphasis to qual
theory & LMs for first quant.
Engineers have always written on napkins & scratched
out bounding calcs on back of envelopes.
Real world of engineering thought is dressed up in
scientific garb until unrecognizable.
Doing so is disservice & misleading.
Road to tech vision is paved with bricks of effective
creative modeling.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 26