1 / 32

Chapter 9

Chapter 9. NONLINEARITIES AND APPROXIMATION. 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. .

salim
Télécharger la présentation

Chapter 9

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 9 NONLINEARITIES AND APPROXIMATION.

  2. MAD Models • Minimize the sum of absolute deviations • The deviation is defined as ei = Yi - ∑XjiBj • 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.

  3. 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.

  4. 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.

  5. Example from Book Predict the price based on quantity of oranges and quantity of juice sold.

  6. Orange Price Model • Pricei = Bo + B1QOrangesi + B2QJuicei + 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.

  7. The LP from this Problem 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.

  8. Variation • Minimize the biggest absolute deviation from the line. • This can be accomplished by using inequality constraints and one e. • Yi - ∑XjiBj ≤ e (if max e is positive) • ∑XjiBj – Yi ≤ e (if max e is negative) • Rewrite these so that Yi is on the RHS and e is on the left. • Resolve our orange example, but restrict b1 to be non-positive and b2 to be non-negative.

  9. Minimize the Largest Absolute Deviation

  10. Optimizing a Fraction McCarl and Spreen Section 9.1.3

  11. Type of Problem Co + CjXj MAX do + djXj S.T. aijXj le bi Xj ge 0 and the denominator strictly positive

  12. Define Some New Variables Yo = [do +djXj]-1 Yj = YoXj (Xj = Yj/Yo)

  13. Do Some Math And . . . Max CoYo + CjYj s.t. -biYo + aijYj le 0 doYo + djYj = 1 Yo, Yj GE 0 Solve this and do the reverse transformation to get the X values

  14. Example Problem 1.8 X1 + 1.7 X2 MAX 10 + 4X1 + 4.1 X2 s.t. 1.5X1 + X2 le 6 3.0X1 + 4X2 le 20 X1, X2 ge 0

  15. Transformed Max 1.8 Y1 + 1.7Y2 s.t -6Yo + 1.5Y1 + Y2 le 0 -20Yo + 3Y1 + 4Y2 le 0 10Yo + 4Y1 + 4.1Y2 = 1 Yo, Y1, Y2 ge 0

  16. Solution X1 = Y1/Yo = 1 1/3 X2 = Y2/Yo = 4

  17. Separable Programming In some cases, the level of return to an activity is not constant as the activity level increases. Because of the law of diminishing returns, returns to additional units of an enterprise often decrease as more units are added.

  18. Diminishing Returns profit from activity j level of activity j

  19. Linear Approximation profit from activity j A B level of activity j

  20. Modeling this Situation Max Profit = R1X1 + R2X2 + R3X3 X1 LE A X2 LE (B-A) And X2 and X3 use the same (or higher) levels of all resources. Because R1 > R2 > R3, X1 will enter solution to its full level A before X2 is selected and so on.

  21. Example

  22. Tableau Totals: A-635; B-100, C-50

  23. Alternative Method This method of solving separable programming is sometimes called the "delta method." An alternative, presented in McCarl and Spreen chapter 9, is sometimes called the "lambda method" or "grid point approximation."

  24. To Solve Our Problem This Way

  25. Tableau Under Lambda Method

  26. McCarl and Spreen McCarl and Spreen present a different application of the gridpoint method in chapter 9, section 2.1. Also, they show you how to extend this method to functions of more than one variable. The more complicated material won't appear on the final but you may want to read it on your own.

  27. Using Integers to Approximate Nonlinear Functions McCarl and Spreen, 15.1.6 We will come back to this material. If you have already seen integer programming, you may want to look at it now.

  28. Some Functions Some functions cannot be approximated using the delta or lambda methods because the more "attractive" section of the function takes place at larger numbers. If, for example, unit profits increase as output increases or costs (in a cost min) decrease as a variable gets larger, those techniques don't work.

  29. Modifying the Lambda Method for a max, P0<P1<P1<P3 The Zi are 0-1 integers

  30. An Example Modification of our original 3 product situation so that the profit from A increases as output goes up.

  31. Without Integers We are skipping the intermediate steps. Profit from 635 units of A is $6200, not $6254.7 as it would appear from this answer.

  32. With the Integer Constraints Profit from the first activity is now calculated correctly.

More Related