Using simplex method solve the following L.P.P Max. $Z = 4x_1 +3x_2 + 6x

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written 7.8 years ago by

teamques10 ★ 70k

• modified 7.6 years ago

We express the given problem in standard form ,taking slack variables

$ s_1, s_2, s_3$

$z – 4x_1 – 3x_2 – 6x_3 + 0s_1 +0s_2+ 0s_3 = 0\\ 2x_1 + 3x_2 + 2x_3 +s_1 +0s_2+0s_3 = 440 \\ 4x_1 +0x_2 +3x_3 +0s_1 +s_2+0s_3 = 470 \\ 2x_1 + 5x_2+0x_3+0s_1 +0s_2+s_3 = 430$

Now we form the simplex table as

Iteration no.	Basic variables	$\frac{coefficients of}{x_1,x_2,x_3,s_1,s_2,s_3}$	R.H.S solution	Ratio
0	z	-4, -3, -6, 0, 0, 0	0
$s_2$ leaves	$s_1$	2, 3, 2, 1, 0, 0	440	440/2=220
$x_3$ enters	$s_2$	4, 0, 3*, 0, 1, 0	470	470/3=156.67
	$s_3$	2, 5, 0, 0, 0, 1	430	430/0 = -
1	z	4, -3, 0, 0, 2, 0	940
$s_1$ leaves	$s_1$	-2/3, 3*, 0, 1, - 2/3, 0	380/3	380/9=42.2
$x_2$ enters	$x_3$	4/3, 0, 1, 0, 1/3, 0	470/3
	$s_3$	2, 5, 0, 0, 0, 1	430	430/5=86
2	z	10/3, 0, 0, 1, 4/3, 0	3200/3
	$x_2$	-2/9, 1, 0, 1/3, -2/9, 0	380/9
	$x_3$	4/3, 0, 1, 0, 1/3, 0	470/3
	$s_3$	28/9, 0, 0, -5/3, 10/9, 1	1970/9

Since all the elements in the row of z becomes positive ,we get solution from R.H.S.

$x_2=380/9 , x_3 = 470/3 , x_1=0 , Z_{Max} = 3200/3 $

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