MTV 13: Squeezing Little Models
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MTV 13: Squeezing Little Models

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Module 13 from Creative Modeling for Tech Vision. Picks up on Little Models theme of module 11 and shows how to get more out of first efforts by "squeezing" the little model to learn more.

Module 13 from Creative Modeling for Tech Vision. Picks up on Little Models theme of module 11 and shows how to get more out of first efforts by "squeezing" the little model to learn more.

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MTV 13: Squeezing Little Models MTV 13: Squeezing Little Models Presentation Transcript

  • Creative Modeling for Technology Visionaries Qualitative & Simplified Quantitative Modeling for Product Creation Module 13: Squeezing Little Models David E. Goldberg University of Illinois at Urbana-Champaign Urbana, Illinois 61801 deg@uiuc.edu
  • Squeezing Little Models Would 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. 2
  • This Module Review 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. 3
  • 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. 4
  • 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. 5
  • Consider Turning Point Derivation Tdecide = T1n Tdo = T2/n Set 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. 6
  • A Specific Case Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 7
  • 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. 8
  • 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: τ & ν. Speed-up representation. Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 9
  • Non-Dimensionalizing Optimal team size: Optimal time: Plug into deciding-doing equation: Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 10
  • Dimensionless Form Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 11
  • Aside: Power Laws & log-log Power 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. 12
  • Speedup as Dimensionless Recasting Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 13
  • 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. 14
  • An Illustrative Example Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 15
  • Time and Quality Relationship For 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. 16
  • 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. 17
  • Nonlinearity in Deciding & Doing Pairwise communications More generally, Likewise, Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 18
  • A Power-Law Model For , monotonically decreases with increased Complexity of decision making increases, the efficient team size decreases. Likewise, monotonically increases with less shirking or more synergy reduces the efficient team size. Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 19
  • Power-Law Solution n: T: Dimensionless form: Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 20
  • 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. 21
  • Same Form as Deciding/Doing Substitute T2 = Tr m and T2 = Td we are back to Similar reasoning can be used to look at flat mTr organizations. n* = Td Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 22
  • 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. 23
  • 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. 24
  • Extensions of EOP Multiplicative 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. 25
  • Some examples of explicit sEOPs Power law: Exponential: Logarithmic: Mixed forms occasionally have explicit solution. Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 26
  • Exponential-Exponential Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 27
  • Power Law of Invertible Function Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 28
  • 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. 29
  • Bottom Line Can 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? Creative Modeling for Tech Visionaries © 2007 David E. Goldberg. All rights reserved. 30