Lecture 4 Linear Programming Models: Standard Form
Lecture 4Linear Programming Models: Standard FormAugust 31, 2009Lecture 4Outline: Standard form LP Transforming the LP problem to Standard form Basic solutions of Standard LP problemOperations Research Methods1Lecture 4Why Standard Form? The simplex method had proven to be the most efficient (practical)solver of LP problems The implementation of simplex method requires the LP problem instandard formOperations Research Methods2Lecture 4What is the Standard Form It is the LP model with the specific form of theconstraints:max (or min)z=c1x1+c2x2+ cnxnsubject toa11x1+a12x2+ +a1nxn=b1a21x1+a22x2+ +a2nxn= +am2x2+ +amnxn=bmx1 0, x2 0, . . . , xn 0 mequalities andnnonnegativity constraints withm nOperations Research Methods3Lecture 4Bringing an LP to its Standard Form The inequality Introduce asurplus variable The inequality Introduce aslack variableNOTE: Thecost of surplus and slack variables is zero Unrestricted variable in sign:Replace it with adifference of two new variables All new variables have to benonnegativeOperations Research M
Lecture 4 What are the basic solutions? • For a problem in the standard form a basic solution is a point ¯x = (¯x1,...,¯x n) that has at least n − m coordinates equal to 0, and satisfies all the equality constraints of the problem a11x¯1 + a12¯x2 + ··· + a1n¯x n = b1 a21x¯1 + a22¯x2 + ··· + a2n¯x n = b2 a m1¯x1 + a m2x¯2 + ··· + a mn¯x n = b m • If the point ¯x has ...
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