1.

Solve by Simplex Method:\[ \operatorname{Max} Z=2 x_{1}+x_{2} \]Subject to \[ \begin{array}{c} 3 x_{1}+x_{2} \leq 3 \\ x_{1}+3 x_{2} \leq 3 \\ x_{1}, x_{2} \geq 0 \end{array} \]

Answer»

Objective function is Max z = 2x1 + x2

By changing given inequalities into equations

3x1 + x2 + x3 = 3

x1 + 3x2 + x4 = 3

x1, x2, x3, x4 \(\geq\) 0

First simplex table

Cj2100Ratio
XBCBX1X2X3X4b
X303*11031→
X40130133
Zj - Cj-2\(\uparrow\)-100

Second simplex table

Cj21100Ratio
XBCBX1X2X3X4b
X1211/31/3013
X4008/3* -1/3126/8 = 3/4 →
Zj - Cj0-1/3 \(\uparrow\)2/30

Third simplex table

Cj2100Ratio
XBCBX1X2X3X4b
X12103/8-1/83/4
X2101-1/83/83/4
Zj - Cj005/81/8


\(\because\) All Zj - Cj \(\geq\) 0

\(\therefore\) x1 = 3/4, x2 = 3/4, x3 = 0, x4 = 0
i.e., x1 = 3/4 and x2 = 3/4 is a solution of given linear programming
\(\therefore\) Max z = 2 x 3/4 + 3/4 = 9/4


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