MTV 13: Squeezing Little Models

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

[email protected]

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.


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.


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.


Do the Math

Take derivative of T(n) with respect to n.

Set to zero.

Do it.

Efficient team size

Optimal time:


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.


A Specific Case


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.


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.


Non-Dimensionalizing

Optimal team size:

Optimal time:

Plug into deciding-doing equation:


Dimensionless Form


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).


Speedup as Dimensionless Recasting


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


An Illustrative Example


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.


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.


Nonlinearity in Deciding & Doing

Pairwise communications

More generally,

Likewise,


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.


Power-Law Solution

n:

T:

Dimensionless form:


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


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 *


Generalizing Little Models

Elementary optimization problems (EOPs).

Elementary turning points (ETPs).

Some solvable classes.

Helps to know what to look for.


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.


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.


Some examples of explicit sEOPs

Power law:

Exponential:

Logarithmic:

Mixed forms occasionally have explicit solution.


Exponential-Exponential


Power Law of Invertible Function


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.


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?

Education

MTV 13: Squeezing Little Models