Answered: Use two-phase simplex Method to solve…

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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Use the simplex method to find x, 20 and x, 2 0 such that Xq + 4x2 2 26 Xq + X2546 and such that z = 12x, + 10x, is maximized. The maximum is z= when x1 and x2 %3D
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minimize subject to x1 + 3x2 - Ax3 — 5x4 − - x1 + x2 x3 + 4x4 = -1 2x1 - X2 - - 3x3 - x4 В x1, x2, x3, x4 ≥ 0 (a) Find all values of ẞ for which the LPP has a BFS x* whose basic variables are x3 and x4. Express x* is terms of ẞ: x* = - (b) For the x* you found in (a), find the reduced cost č; = c; — c—B˜¹A; for the basic direction j = 1.
Minimize Z = (-1)*(X, – 2X2 + 3X3) Subjected to, X, + X, + X357 -X, + X2 – X3 < -2 3X, – X2 - 2X3 = -5 X,, X, 2 0 2
2. Minimize z - 6х, + 6х, + 8x, + 9х, subject to X, + 2xz + x3+ X4 2 3 2x, + xz + 4x3 + 9x42 8 X, 2 0, x2 2 0, x3 2 0, x,2 0.
Minimize c = 3x + 3y subject to x + 2y ≥ 38 2x + y ≥ 38 x ≥ 0, y ≥ 0.
Minimize: F = (xỉ + 4x3 – 8x, – 16x2) Subject to: X1 + x2 < 5 0
Find the optimal solution: Zmax = 8x1 + 7x2 Subject to: 3x1 + X2 s 66 X1 + X2 s 45 X1 s 20, x2 s 40 X1, X2 2 0
Sketch the following problem graphically: max f = x2 X2 subject to g1 = X1 – (1 – x2)³ < 0 92 = x1 2 0 Find the solution graphically. Apply the optimality conditions and monotonicity rules. Discuss. (From Kuhn & Tucker, 1951.)
Question 29 Maximize 2x1 + 4x2 subject to x1 + 3x₂ �� 12 2x1 + x₂ ≤ 14 120 and ₂ > 0 Using simplex method by introducing slack variables 3 and 4, and M = 2x1 + 4x2, now the inequalities are changed to equations x1 + 3x2 + x3 = 12 (1) 2x1 + x₂ + x4 = 14 (2) 2x14x2 M = 0 (3) To bring 1 and 2 into the solution, which one (1 or 2) is firstly brought into the solution and which equation is used as the pivot equation. O1 and (2) O #2 and (3) O2 and (2) O #1 and (1) O2 and (1)
Solve the following problems by using the Simplex Method and verify the solution graphically. * 8.43 Minimize f=9x₁ + 2x2 + 3x3 subject to 2x₁ + x₂-3x3-5 X1-2x2 + 2x3 = -2 X1, X2, X3 ≥ 0 8.46 Minimize f= x₁-x₂ subject to 4x₁ + 3x₂ ≤ 12 x₁ + 2x₂ ≤ 4 4 ≥ 2x₁ + x₂ X1, X₂ ≥ 0 8.47 Maximize z = 2x1 + 3x₂ subject to x₁ + x₂ ≤ 16 -X₁-2x₂-28 24 ≥ 2x₁ + x2 X1, X2 ≥ 0
Use the Dual Simplex method (show steps) to solve this LP problem: Min z = 2x₁ + 10x2 + 3x3 2x1 + x₂ + x3 s.t. 8x₁ - x₂ + 4x3 X1, X2, X3 20 > 20 < 10

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Use two-phase simplex Method to solve the following LP Minimize z= 6 x1 + 3 x2 Subject to x1+x2 ≥ 1 2x1- x2 ≥ 1 3x2 ≤ 2 x1,x2 ≥ 0