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lecture10 - Soft Constraints IE426 Optimization Models and...

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IE426: Optimization Models and Applications: Lecture 10 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University October 3, 2006 Jeff Linderoth IE426:Lecture 10 Soft Constraints Suppose I “kind of” want a constraint to hold. This happens A LOT in the real world One way to enforce “kind of” is to allow the constraint to be violated, but at a price. j N a j x j b j N a j x j - u b , u 0 Add a “penalty” term λu to the objective This term should be: + if min, - if max Jeff Linderoth IE426:Lecture 10 Any Constraint can be “Softened” j N a j x j b j N a j x j - u b , u 0 Add a “penalty” term λu to the objective + if min, - if max j N a j x j b j N a j x j + v b , v 0 Add a “penalty” term λv to the objective + if min, - if max j N a j x j = b j N a j x j - u + v b , u 0 , v 0 Add a “penalty” term λ ( u + v ) to the objective Jeff Linderoth IE426:Lecture 10 A New Example. Smallville, USA Smallville is a city of 20,000 inhabitants. The city council is in the process of developing an equitable city rate tax table. Taxes come from a combination of four sources Property taxes: ($550M base) Food & Drugs: ($35M base) Other Sales: ($55M base) Gasoline: (Consumption: 7.5 million gallons/year) Smallville would like to come up with a “fair” city tax... Tax revenues must be at least $16M The property tax rate should be 1%. Food and drug taxes must be 10% of all the taxes collected Sales taxes cannot exceed 20% of all the taxes collected The gasoline tax must be $.02/gallon. Jeff Linderoth IE426:Lecture 10
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Smallville, Cont. Decision Variables r p Property tax rate r f Food/Drug tax rate r s Sales tax rate r g Gasoline tax rate in $/gallon x p Property tax collected (in $) x f Food/Drug collected (in $) x s Sales tax collected (in $) x g Gasoline tax collected (in $) T Taxes Collected (in $) Taxes (from rates) are based on the tax base Jeff Linderoth IE426:Lecture 10 SmallVille—Scaled min e p + e f + e s + e g s.t.
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