mtv 13: squeezing little models
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Creative Modeling forTechnology VisionariesQualitative & Simplified Quantitative Modeling for Product Creation
Module 13: Squeezing Little Models
David E. GoldbergUniversity of Illinois at Urbana-ChampaignUrbana, Illinois 61801
deg@uiuc.edu
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.2
Squeezing Little ModelsWould like to move from qual to quant in difficult domains.
Once model obtained, how to we improve it, squeeze it, and extend it?
In this way, little modeling begets more little modeling.
Want to squeeze the most out of little models possible.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.3
This ModuleReview deciding-doing model:
Beyond deciding-doing:Stretching: auxiliary models.
Modifying: modification to functional form.
Reusing: Same math, different app.
Generalizing little models: EOPs and ETPs
Some solvable classes.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.4
Deciding and Doing Model
Team size: n
Discussing what is to be done: T1
Total time to do the task alone: T2
Total time required for task completion:
Model integration via summation.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.5
Do the Math
Take derivative of T(n) with respect to n.
Set to zero.
Do it.
Efficient team size
Optimal time:
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.6
Consider Turning Point DerivationTdecide = T1n
Tdo = T2/nSet equal to each other.
T1n* = T2/n*Same as before:
Not generally the case, but not bad approximation.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.7
A Specific Case
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Stretching a Model
Ways to stretch a model.
Dimensional analysis can help reveal essential form.
Can recast in useful terms.
Can add auxiliary models to basic model.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.9
Ratios Reveal Structure
Can non-dimensionalize with respect to optimal solutions.
Can non-dimensionalize with respect to meaningful benchmarks.
Deciding-doing examples:Non-dimensional representation: & ν.
Speed-up representation.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.10
Non-Dimensionalizing
Optimal team size:
Optimal time:
Plug into deciding-doing equation:
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.11
Dimensionless Form
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.12
Aside: Power Laws & log-logPower law: f(x) = axb
Consider log-log transformation.
ln f(x) = ln axb = ln a + blnx
log-log transforms power law to linear curve with slope b (and intercept lna).
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.13
Speedup as Dimensionless Recasting
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Auxiliary Models: Decision Quality
Beyond efficiency: Quality
Solutions successfully proposed by individual team members with a probability p.
Solution quality:
Q increases monotonically with increased n.
Q is high when
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.15
An Illustrative Example
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Time and Quality RelationshipFor n<n*, longer completion and lower solution quality.
For n>n*, a better solution quality is achieved in exchange for longer completion time.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.17
Modifying a Little Model
What if deciding and doing are not linear & hyperbolic respectively?
Can modify the form.
For example, imagine that pairwise interactions are important in decision.
Deciding might be quadratic function of n.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.18
Nonlinearity in Deciding & Doing
Pairwise communications
More generally,
Likewise,
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.19
A Power-Law Model
For , monotonically decreases with increased
Complexity of decision making increases, the efficient team size decreases.
Likewise, monotonically increases withless shirking or more synergy reduces the efficient team size.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.20
Power-Law Solution
n:
T:
Dimensionless form:
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.21
Reusing Little Models
Consider problem of sizing breakouts in a conference (IlliGAL 2005021).
Have a big meeting of size m.
Have k = m/n breakout groups of size n.
Incur time of discussion Td per member in breakout groups and Tr reporting per team.
T = Td n + Tr k = Td n + Tr m/n
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.22
Same Form as Deciding/DoingSubstitute T2 = Tr m and T2 = Td we are back to
Similar reasoning can be used to look at flat organizations.
d
r
T
mTn *
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.23
Generalizing Little Models
Elementary optimization problems (EOPs).
Elementary turning points (ETPs).
Some solvable classes.
Helps to know what to look for.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.24
Elementary Optimization Problems
Elementary optimization problems (EOPs). A function of one variable.
The sum of a monotonically increasing function and a monotonically decreasing function.
Twice differentiable.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.25
Extensions of EOPMultiplicative form
Explicit EOP and implicit EOP
Single optimum simple EOP (sEOP)
Clearly nothing special about deciding, doing, time, quality, etc.
Have done transaction costs & span of control.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.26
Some examples of explicit sEOPs
Power law:
Exponential:
Logarithmic:
Mixed forms occasionally have explicit solution.
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Exponential-Exponential
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Power Law of Invertible Function
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Similar Argument for Turning Points
Have increasing and decreasing function.
Interested when they are equal.
f(n) = g(n).
Need explicit solution for n for greatest utility in inspection.
Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved.30
Bottom LineCan squeeze a lot out of little models.
Stretch them (visualize effectively and use auxiliary models).
Modify them (add complexity, accuracy).
Reuse them (in other domains).
Generalize them in EOPs and ETPs.
Seeking explicit models that yield qualitative & quantitative insight cheaply.
Can we extend them further & integrate multiple models for complex domains?
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