Chapter 7

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Chapter7TheSimplexMethodInthischapter, ,youwillbeabletoidentifywhenaproblemhasa lternateoptimalsolutions(SOLVERnevertell syouthis:italwaysgiveyouonlyoneoptimalso lution). ,wepointoutthateverylinearprogramcanbeco nvertedinto\standard"formMaxc1x1+c2x2+:: :+cnxnsubjecttoa11x1+a12x2+:::+a1nxn=b1: ::::::::am1x1+am2x2+:::+amnxn=bmx1 0;:::xn 0wheretheobjectiveismaximized, : Iftheproblemisminz,convertittomax z. Ifaconstraintisai1x1+ai2x2+:::+ainxn bi, +ai2x2+:::+ainxn+si=bi,wheresi 0. Ifaconstraintisai1x1+ai2x2+:::+ainxn bi, +ai2x2+:::+ainxn si=bi,wheresi 0. Ifsomevariablexjisunrestrictedinsign,rep laceiteverywhereintheformulationbyx0j x00j,wherex0j 0andx00j 2x1+3x2x1 3x2+2x3 3 x1+2x2 2x1urs;x2 0;x3 0Letus 3x2x1 3x2+2x3+s1=3 x1+2x2 s2=2x1urs;x2 0;x3 0s1 0s2 0Thelaststepistoconverttheunrestrictedva riablex1intotwononnegativevariables:x1=x 01 2x001 3x2x01 x001 3x2+2x3+s1=3 x01+x001+2x2 s2=2x01 0;x001 0;x2 0;x3 0s1 0s2 ,inthiscoursewesolve\byhand"onlythecasew heretheconstraintsareoftheform +x22x1+x2 4x1+2x2 3x1 0;x2 0First,weconverttheproblemintostandardfo rmbyaddingslackvariablesx3 0andx4 +x22x1+x2+x3=4x1+2x2+x4=3x1 0;x2 0

Chapter 7 The Simplex Metho d In this c hapter, y ou will learn ho w to solv e linear programs. This will giv ey ou insigh ts in to what SOL VER and other commercial linear programming soft

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