SM233 Spring 2006 Part 1: Optimization Models

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SM233 Spring 2006Part 1 Optimization Models

• 2 Multi-Variable Optimization
• 2.2 Constrained Optimization

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Assignment

• Due
• Problems p. 54 1b), c) d) 5c)d)
• Due Friday, 10 February 2006
• Pending
• Matlab Lab assignment 4
• Due Tuesday, 14 February 2006
• Assign
• Problems p. 54 2a), 6a), 10a)
• Due Wednesday, 22 February 2006
• Exam 1 Tuesday, 14 February 2006

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Points to Control

• Constrained multi-variable optimization problems
• The Mechanics of Modeling
• The five-step method where constraints arise
• Methods of optimization
• Reprise 2D/3D constrained max/min problems
• Reprise Lagrange multiplier
• Computational tools
• Maple (Symbolic)
• MatLab (Numerical)

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Models and design

• The design paradigm

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Context constrained optimization

• Example 2.2 Here, TV, TV, but limited
  production capacity
• 19in TVs base price 339/set
• Cost to manufacture 195/set
• 21in TVs base price 399/set
• Cost to manufacture 225
• Market incentives
• 19in drop base price 0.01_at_19inSet sold
  0.003_at_21inSet sold
• 21in drop base price 0.01_at_21inSet sold
  0.004_at_19inSet sold
• Fixed costs 400,000 overhead
• Production constraints
• 5000 max of 19in TVs/yr than can be produced
  (parts limit)
• 8000 max of 21in TVs/yr than can be produced
  (parts limit)
• 10,000 max of all TVs/yr the factory can
  produce (size limit)
• Seek for max profit
• How many sets of each type should be manufactured
  ?
• (presuming all manufactured are sold)
• What would be the max profit?

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The model

• Variables, constants
• s 19in sets sold (per yr) independent
• t 21in sets sold (per yr) independent
• p selling price for 19in sets () p p(s,t)
• q selling price for 21in sets () q q(s,t)
• C cost of manufacturing sets (/yr) C(s,t)
• O overhead () constant
• R total revenues (/yr) R R(p,q,s,t)
  R(s,t)
• P profit (/yr) P P(R,C)

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The model

• Constitutive relations (5 for 5)
• Constraints on independent variables

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The model

• Seek
• Maximize P(s,t) with respect to (s,t)

over the range of values compatable with the
constraints (feasible values)
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Mathematical formulationresoluton

• Objective function
• Maximize P(s,t) over the (constrained) domain
• Resolution HOW?

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Pure math toolsconstrained optimization

• Problem (calc 3) Maximize f(x,y) x2 - y2 over
  the curve
• x2 y2 1
• See the problem ( lecture02_2_append.mw )
• Problem (Ex. 2.3, p.37)
• Maximize f(x,y,z) x - 2y 3z over the 2D
  surface
• x2 y2 z2 3
• See the problem ( lecture02_2_append.mw )
• Abstract Maximize f(x1,x2,,xn) over the (n-1)D
  surface (of constraint) g(x1,x2,,xn) c .
  (Here, n 2, 3)

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The math problem its resolution

• The problem maximize over
  the profit fn
• Resolution in principle, construct
• Solve for
• Compute mat profit

1st deriv. Test
(2nd deriv. Test for max pt)
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The math problem implementing the resolution

• Analytical approach Maple (Mathematica)
• Numerical approach (see chapter 3) MatLab
• This problem Section02.1_append.mw
• MatLab see logSession02_1.txt (load
  data02_2.mat)

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Sensitivity analysis

• p. 26 How sensitive are sopt , topt the
• the price elasticity for 19in sets?
• The extended model
• Variables and parameters
• a price elasticity of 19in sets (/set) a
  0.01
• Constitutive relations Objective function

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Sensitivity analysis the functions

• The problem maximize over
  the profit fn
• Resolution in principle, construct
• Solve for

1st deriv. Test
(2nd deriv. Test for max pt)
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Sensitivity to parametersrelative measure

• Defn (p.12). Sensitivity of sopt to a
• Relative change sopt to a, per unit ratio of sopt
  to a
• In principle

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The math problem implementing the resolution

• Analytical approach Maple (Mathematica)
• Numerical approach (see chapter 3) MatLab
• This problem Section02.1_append.mw
• MatLab see logSession02_1.txt (load
  data02_1.mat)

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