Optimization Please solve the attached minimization problem manually Objective function:subject to:andMin:Z = 2 x₁ - 3 X₂-4 X3X₁ +5x₂-3x3 ≤ 15X₁ + X₂ X3 ≤ 115 X₁-6x₂ + x3 ≤ 4X1 X2 X3 20

Answered: Objective function: Min: Z= 2 x₁ -…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

Optimization Please solve the attached minimization problem manually

Objective function:
subject to:
and
Min:
Z = 2 x₁ - 3 X₂-4 X3
X₁ +
5x₂-3x3 ≤ 15
X₁ + X₂ X3 ≤ 11
5 X₁-6x₂ + x3 ≤ 4
X1 X2 X3 20

Expert Solution

Step 1

Use simplex method to solve the minimization problem.

Convert the problem into canonical form using slack variables.

Min $Z = 2 x_{1} - 3 x_{2} - 4 x_{3} + 0 S_{1} + 0 S_{2} + 0 S_{3}$

subject to

$x_{1} + 5 x_{2} - 3 x_{3} + S_{1} = 15 x_{1} + x_{2} + x_{3} + S_{2} = 11 5 x_{1} - 6 x_{2} + x_{3} + S_{3} = 4 x_{1}, x_{2}, x_{3}, S_{1}, S_{2}, S_{3} \geq 0$

Iteration 1		$C_{j}$	2	-3	-4	0	0	0
$B$	$C_{B}$	$X_{B}$	$x_{1}$	$x_{2}$	$x_{3}$	$S_{1}$	$S_{2}$	$S_{3}$	min ratio $\frac{X_{B}}{x_{3}}$
$S_{1}$	0	15	1	5	-3	1	0	0
$S_{2}$	0	11	1	1	1	0	1	0	$\frac{11}{1} = 11$
$S_{3}$	0	4	5	-6	1	0	0	1	$\frac{4}{1} = 4$
$Z = 0$		$Z_{j}$	0	0	0	0	0	0
		$Z_{j} - C_{j}$	-2	3	4	0	0	0