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Unformatted text preview: LP not in standard form is: max z = 4x1 + 3x2 s.t. x1 + x2 ≤ 40 2x1 + x2 ≤ 60 x1, x2 ≥ 0 (leather constraint) (labor constraint) SoluWon: x1=20, x2=20, z = 140 The same LP in standard form is: max z = 4x1 + 3x2 s.t. x1 + x2 + s1 = 40 2x1 + x2 + s2 = 60 x1, x2, s1, s2 ≥ 0 SoluWon: x1=20, x2=20, s1=0, s2=0, z = 140 In summary, if a constraint i of an LP is a ≤ constraint, convert it to an equality constraint by adding a slack variable si to the ith constraint and adding the sign restricWon si ≥ 0. 4.1 – How to Convert an LP to Standard Form A ≥ constraint can be converted to an equality constraint. Consider the formulaWon below: min z = 50 x1 + 20x2 + 30X2 + 80 x4 s.t. 400x1 + 200x2 + 150 x3 + 500x4 ≥ 500 3x1 + 2x2 ≥ 6 2x1 + 2x2 + 4x3 + 4x4 ≥ 10 2x1 + 4x2 + x3 + 5x4 ≥ 8 x1, x2, x3, x4 ≥ 0 To convert the ith ≥ constraint to an equality constraint, deﬁne an excess variable (someWmes called a surplus variable) ei (ei will always be the excess variable for the ith ≥ constraint. We deﬁne ei to be the amount by which ith constraint is over saWsﬁed. 4.1 – How to Convert an LP to Standard Form Transforming the LP on the previous slide to standard form yields: min z = 50 x1 + 20x2 + 30X2 + 80 x4 s.t. 400x1 + 200x2 + 150 x3 + 500x4 – e1 = 500 3x1 + 2x2 ‐ e2 = 6 2x1 + 2x2 + 4x3 + 4x4 ‐ e3 = 10 2x1 + 4x2 + x3 + 5x4 ‐ e4 = 8 xi, ei > 0 (i = 1,2,3,4) In summary, if the i th constraint of an LP is a ≥ constraint, it can be converted to an equality constraint by subtracWng the excess variable ei from...
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- Fall '11