mccarl-chapter9

# mccarl-chapter9 - Chapter 9 NONLINEARITIES AND...

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Chapter 9 NONLINEARITIES AND APPROXIMATION.

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MAD Models Minimize the sum of absolute deviations • The deviation is defined as ei = Yi - ∑X ji B j Similar to regression, except we minimize sum of the absolute values of the error term and not the sum of the squares of the error terms.
Problem Defined Minimize ∑ | ei | Subject to ei - ∑ Xji bj = Yi The bj are the variables to be determined in the model. They can be positive or negative, as can the ei.

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Need Substitutions to solve Define ei = eipos – eineg Define bj = bjpos – bjneg If we replace these in the formulation, we can solve these. Remembering that the absolute value is always a positive number, we have an objective function that sums up over all eipos + eineg.
Example from Book Price of Oranges Quantity of Oranges Sold Quantity of Juice Sold 10 8 5 5 9 1 4 10 9 2 13 8 6 15 2 9 17 3 Predict the price based on quantity of oranges and quantity of juice sold.

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Orange Price Model • Price i = Bo + B1QOranges i + B2QJuice i + e i The ei can be positive or negative, as can the Bj. Decision variables are the Bj and the ei which are determined by the choice of the Bj. Because I minimize the sum of absolute values, I must define two error terms for each observation, the positive and the negative.
The LP from this Problem e1p e1n e2p e2n e3p e3n e4p e4n e5p e5n e6p e6n b0 b1 b2 min RHS used Obj 1 1 1 1 1 1 1 1 1 1 1 1 11.277 obs 1 1 -1 1 8 5 eq 10 10 2 1 -1 1 9 1 eq 5 5 3 1 -1 1 10 9 eq 4 4 4 1 -1 1 13 8 eq 2 2 5 1 -1 1 15 2 eq 6 6 6 1 -1 1 17 3 eq 9 9 answers 5.79 0 0 0 0 0 0 2.72 0 0 2.77 0 3.43 0.19 -0.15 Here, I've let the Bj be unrestricted in sign, because Solver will let me do that. I could add more columns and define negative Bj.

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Variation Minimize the biggest absolute deviation from the line. This can be accomplished by using inequality
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## This note was uploaded on 11/15/2011 for the course AGEC 7100 taught by Professor Duffy,p during the Fall '08 term at Auburn University.

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mccarl-chapter9 - Chapter 9 NONLINEARITIES AND...

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