Chapter 7

Chapter7 TheSimplexMethodInthischapter, ,youwillbeabletoidentifywhenaproblemhasa lternateoptimalsolutions(SOLVER nevertellsyouthis:italwaysgiveyouonlyone optimalsolution). ,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 0 Letus 3x2x1 3x2+2x3+s1=3 x1+2x2 s2=2x1urs;x2 0;x3 0s1 0s2 0 Thelaststepistoconverttheunrestrictedvar iablex1intotwononnegativevariables:x1=x0 1 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 0 First,weconverttheproblemintostandardfor mbyaddingslackvariablesx3 0andx4 +x22x1+x2+x3=4x1+2x2+x4=3x1 0;x2 0x3 0;x4 ,z=x1+x2or,equivalently,z x1 x2=0:Puttingthisequationtogetherwiththec onstraints, x1 x2=0 Row02x1+x2+x3=4 Row1x1+2x2+x4=3 Row2( ) ,whilesatisfyingtheseequationsand,inaddi tion,x1 0,x2 0,x3 0,x4 ,theequationsaresolvedintermsofthenonbas icvariablesx1, (otherthanthespecialvariablez) :Isthisanoptimalsolutionorc

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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