Lecture 5

# Thefromofsuchanlpis maxorminzc1x1c2x2cnxn st

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Unformatted text preview: the i th constraint and adding the sign restricWon ei ≥ 0. 4.1 – How to Convert an LP to Standard Form If an LP has both ≤ and ≥ constraints, apply the previous procedures to the individual constraints. Consider the example below. Nonstandard Form Standard Form max z = 20x1 + 15x2 max z = 20x1 + 15x2 s.t. x1 ≤ 100 s.t. x1 + s1 x2 ≤ 100 = 100 x2 + s2 = 100 50x1 + 35x2 ≤ 6000 50x1 + 35x2 + s3 = 6000 20x1 + 15x2 ≥ 2000 20x1 + 15x2 x1, x2 > 0 ‐ e4 = 2000 x1, x2, s1, s2, s3, e4 > 0 4.2 – Preview of the Simplex Algorithm Suppose an LP with m constraints and n variables has been converted into standard form. The from of such an LP is: max ( or min) z = c1x1 + c2x2 + … +cnxn s.t. a11x1 + a12x2 + … + a1nxn =b1 a21x1 + a22x2 + … + a2nxn =b2 . . . . . . am1x1 + am2x2 + … + amnxn =bm xi ≥ 0 ( i = 1,2, …, n) 4.2 – Preview of the Simplex Algorithm If we deﬁne: a11 a21 A .... a m1 a x1 a .... a x2 22 2n x .... .... .... .... x a .... a m2 mn n 12 .... a 1n b1 b2 b .... b m The constraints may be wriaen as a system of equaWons Ax = b. Consider a system Ax = b of m linear equaWons in n variables (where n ≥ m). A basic soluCon to Ax = b is obtained by sebng n – m variables equal to 0 and solving for the remaining m variables. This assumes that the columns for the remaining m variables are linearly independent – (matrix rank = m). To ﬁnd a basic solu1on to Ax = b, we choose a set of n – m variables (the nonbasic variables, or NBV) and set each of these variables equal to 0. Then w...
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