SM233 Spring 2006 Part 1: Optimization Models

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SM233 Spring 2006Part 1 Optimization Models

• 1 One Variable Optimization
• 1.2 Sensitivity Analysis

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Assignment

• Due
• Problems, p. 17 1a), 3a)b), 5a), 8a)
• Due Tuesday, 17 Jan. 2006 UPDATE Friday 20 Jan.
  2006
• Pending
• Matlab Lab assignment 2
• Due Tuesday, 24 January 2006
• Assign
• Problems, p. 17 1b), 3c), 5b), 8b)
• Due Friday, 27 January 2006

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Points to Control

• The sensitivity of a model with respect to
  parameters
• What is sensitivity?
• Why is it important?
• The Mechanics of analyzing sensitivity
• The tools
• How we engage them
• What information they provide

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Models and design

• The design paradigm

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Context word problems

• Example 1.1 Here, piggy, piggy,
• 200 lb Pig gains 5 lbs/day, costs .45/ day
• Market for pork .65/lb, falls 0.01/day
• When sell?
• Last time formulate model maximize wrt t
• P P(t) (0.65 rt)(200 gt) - 0.45t
• -grt2 (0.65g - 200r 0.45)t
  130
• for r 0.01/day, g 5 (lb/day)
• Conclude
• Sell at topt 8 days, Pmax P(topt) 133.20

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Complication What if

• But what if
• Price drop , weight gain
• Error but how bad is it?
• Will small change in price rate/weight gain
  produce small or large changes in topt or Pmax?
• E.g. is the model
• insensitive , very sensitive to variations of
  parameters
• How sensitive? Quantify the sensitivity?

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Sensitivity of the modeltools for analysis

• Embed the model into a family of models
• Objective function
• Sensitivity to r
• Fix g ( 5) GRAPH P(tr). See Pmax(r)
• Tool of analysis slope of Pmax(r) curve
• Sensitivity to g
• Fix r ( 0.01) GRAPH P(tg). See Pmax(g)
• Tool of analysis slope of Pmax(g) curve

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Model variation with parametersforge analytical
tools

• Embed the model into a family of models
  Objective function
• Sensitivity of topt to r, g fixed (g 5)
• gphBest time to sell x vs rate of price drop r,
  (g 5)
• First tool for assessment rate

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Model variation with parametersforge analytical
tools

• Objective function
• Sensitivity of topt to g, r fixed (r 0.01)
• gphBest time to sell x vs rate wt gain g, (r
  0.01)
• First tool for assessment rate

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Sensitivity to parametersrelative measure

• Sensitivity of topt to r, g fixed (g 5) rate
• Problem sensitive to scale (units)
• magnitude cant convey info
• Modify relative rate rate is independent of
  units
• Defn (p.12). Sensitivity of x (e.g. topt) to r
• Relative change x to r, per unit ratio of x to r

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Sensitivity to parametersExample

• Sensitivity of x (e.g. topt) to r, g fixed
  relative rate
• Sensitivity at r 0.01, x(0.01) 8
  (independent of scale)
• A 1 increase in r reduces the best time to
  sell by 3.5

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Sensitivity to parametersExample

• Sensitivity of x (e.g. topt) to g, r fixed
  relative rate
• Sensitivity at g 5, x(5) 8 (independent of
  scale)
• A 1 increase in g increases the best time to
  sell by 3.06

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Sensitivity to parametersContinue

• Sensitivity of profit (e.g. Pmax) to g, r fixed
• Sensitivity of profit (e.g. Pmax) to r, g fixed
• See Homework.

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Summary

• Sensitivity ? relative rate of change of
  optimized quantity to one parameter other
  parameters fixed.
• Focus the mechanics of computing the measure
  (slight interpretation)

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