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Unformatted text preview: METHODS FOR SOLVING LINEAR PROGRAMS I 1 Simplex Method and its Variants. (1947 by George Dantzig) 2 Interior Point Methods  Affine Scaling Method and its Variants. (1964 by Ilya Dikin) 3 interior Point Methods  Projective Transformation Method and its variants. (1984 by Karmarkar) 4 Interior point methods  Barrier Method and its variants. (1986 and onwards, various) IOE 310 Fall 2007 CHAPTER 4  SIMPLEX ALGORITHM I Before we apply the SIMPLEX METHOD, the linear program must be standardized. This form is called the: STANDARD FORM All constraints are equality. All variables are nonnegative. Objective Function may be max or min. IOE 310 Fall 2007 AN EXAMPLE I A company produces two types of belts: deluxe and regular. The data is: deluxe regular available Labor 1 1 40 Leather 2 1 60 Profit $ 4 3 Decision Variables: x 1 = # of deluxe belts. x 2 =# of regular belts. IOE 310 Fall 2007 THE LP MODEL maximize z = 4 x 1 + 3 x 2 x 1 + x 2 40 2 x 1 + x 2 60 x 1 x 2 IOE 310 Fall 2007 For STANDARD FORM add SLACK VARIABLES s 1 = 40 x 1 x 2 s 2 = 60 2 x 1 x 2 LP in standard form: max z 4 x 1 + 3 x 2 = z x 1 + x 2 + s 1 = 40 2 x 1 + x 2 + s 2 = 60 IOE 310 Fall 2007 SOLUTION OF STANDARD MODEL We rewrite the system as: 4 x 1 + 3 x 2 = z x 1 + x 2 + s 1 = 40 2 x 1 + x 2 s 2 = 60 Solution here is x 1 = 0, x 2 = 0, s 1 = 40, s 2 = 60, z = 0. Can this be optimum or can we find a better solution?...
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This note was uploaded on 04/02/2008 for the course IOE 310 taught by Professor Saigal during the Spring '08 term at University of Michigan.
 Spring '08
 Saigal

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