Question
Solve the following LPP by simplex method.
Max z = 3x + 2y
s.t. 2x + y < 18
2x + 3y < 42
3x + y < 24
x, y < 0 .
Answer :
Word Count : 1046
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Skip in 10
Introduce slack variables $s_1, s_2, s_3 \ge 0$ for the inequalities: $$ \begin{aligned} 2x + y + s_1 &= 18 \\ 2x + 3y + s_2 &= 42 \\ 3x + y + s_3 &= 24 \end{aligned} $$ Objective function: Max $z = 3x + 2y \Rightarrow z - 3x - 2y = 0$. Initial simplex tableau: $$ \begin{array}{c|ccccc|c} \text{BV} & x & y & s_1 & s_2 & s_3 & \text{RHS} \\ \hline s_1 & 2 & 1 & 1 & 0 & 0 & 18 \\ s_2 & 2 & 3 & 0 & 1 & 0 & 42 \\ s_3 & 3 & 1 & 0 & 0 & 1 & 24 \\ \hline z & -3 & -2 & 0 & 0 & 0 & 0 \end{array} $$ Step 1: Entering variable: $x$ (most negative in z-row: -3) Leaving variable: compute ratios RHS / pivot column: $$ \frac{18}{2} = 9, \quad \frac{42}{2} = 21, \quad \frac{24}{3} = 8 $$ Smallest positive ratio → $s_3$ leaves. Pivot = 3. Step 2: Pivot on s\_3 row, x-column: New $s_3$ row divided by 3: $$ x + \frac{1}{3}y + \frac{1}{3}s_3 = 8 $$ Update other rows: * Row s\_1: $s_1 - 2 \times \text{new s_3 row}$ $$ 2x + y + s_1 - 2(x + \frac{1}{3}y + \frac{1}{3}s_3) = 18 - 16 \Rightarrow -\frac{1}{3}y + s_1 - \frac{2}{3}s_3 = 2 $$ Multiply by 3: $-y + 3s_1 - 2s_3 = 6 \Rightarrow y - 3s_1 + 2s_3 = -6$ → adjust sign carefully: Better: $s_1 - 2*(x + 1/3 y + 1/3 s_3) + y = s_1 - 2x - 2/3 y - 2/3 s_3 + y + 2x = s_1
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