Q4. Consider the following LP problem and its optimal final tableau shown below:(LP): max z = 2x1 + x2-x3Basic x1 x2 x3 x4 x5 RHSs.t.Z0 3x1 + 2x2 + x3 ≤8-x1 + x2-2x3 ≤4x1121x1, x2, x3 20x50 3-1x4 and x5 are slack variables in the 1s., and 2nd constraints respectively,a) Fill out the empty cells in the given optimal simplex tableau.b)What is the optimality range for c, ?What is the feasibility range for b₂ ?If the coefficient of x2 in the objective function is changed from 1 to 3, calculate thecorresponding changes, detemine whether the current optimum is changed?c)d)1101

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Answered: Q4. Consider the following LP problem…

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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3 Find alternative optimal solutions to the following LP: max z = x₁ + x₂ s.t. X1 Xx₁ + x₂ + x3 ≤ 1 x2 X1 + 2x3 = 1 All x; ≥ 0
Formulate Karush-Kuhn-Tucker conditions for the following task: min –3x + x + 2x3 + 2 (x1 + x2 + x3) s.t. x + x + x = 1
3. Consider the following dual problem Minimize Z = 7x₁ + 2x₂ + 5x3 + 4x4 subject to 2x₁ + 4x₂ + 7x3 + x4 ≥ 5 8x₁ + 4x2 + 6x3 + 4x4 ≥ 8 3x₁ + 8x₂ + x3 + 4x4 ≥ 4 and x₁ ≥ 0, x₂ ≥ 0, X3 ≥ 0, X4 ≥ 0. (a) Construct the primal problem corresponding to this problem. (b) Solve the primal problem by the original simplex method (in tabular form) step by step. Identify the complementary basic solution for the dual problem obtained at each iteration. Use the dual simplex method manually to solve the dual problem. Compare the resulting sequence of basic solutions with the complementary basic solutions obtained in part (b). (c)
Q1. Given the following multi-objective problem (A): max 2X, + 0.5X2 + Xҙ (Z1) (Z2) (Z3) min 4X1 – 3X2 + X3 min X2 – 2X3 s.t. (1) 4X1 + X3 < 10 5X1 + X2 < 20 X; are non-negative Formulate this problem using Global Criterion (GC) method and Compromise Programming (GP) method.
Consider the following LP. Max z = 4x, +8x, +8x, – 6x, subject to X+ X, +X3 %3D X + 4x, +x, = 8 X, X2, X3, X, 20 a. Write the associated dual problem. b. Use the properties of the dual problem to show that the basic solution ( x, ,x, ) is not optimal The optimal solution to the primal LP problem yields the following objective equation: z+2x, + 0x, +0x, + 3x, = 16 Determine the associated optimal dual solution.
An LP solution always provides a lower bound to an IP minimization problem: O True O False W
Example: Given f(x) = 2x - 7 and g(x) = x2 + 4x + 4, find (fog)(x) and (go )(x) no znoto eroltorioo inaiteuD bas toubonponolid arm2 nicoobear Solution: (f g)(x) = f(g(x)) = 2(g(x)) – 7 bonob Replace x by g(x) = 2 (x2 + 4x + 4) – 7 Idiod ot nommo bubog Replace g(x) by x² + 4x + 4 =2x2 + 8x + 1 Bou) Apply Distribution Law and Combine like terms Thus (f°g)(x) = 2x² + 8x + 1 STUDENT PLAYSHOP: WRITTEN WORKS I. As a dependable Columbanite, determine whether each statement is TRUE or FALSE. Show your complete and neat solution to prove your answer. 1. If f(x) = x -4 and g(x) = x + 5, then (f- g)(x) = 2 mua Srit sniursis (8-S ) + ()- (2-+ (x) bas (x) bDA idinoo 2. If f(x) = 5x -4 and g(x) = 3x + 5, then (f+g)(x) =8x + 1 anoitonul lo som b ot sniroto (x) mori (x)g foRnidu ()8-(-(2)(-1).d aast slil onidmo nosmt sdi lodoubog edt onianoto 3. If f(x) = 5x -4 and g(x) = 3x + 5, then (fg)(x) = 5x + 4 (x)ge(x)-(x)() (E- YS+) (S-XS)= less than o (x)8 \(x) = (x) \7.5 II. Find a.) (f•g)(x) and b.) (g °…
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3. Determine whether x = T (0 3 0 9 2) is a solution to 3x2 + 2x5 max X1,2,3,4,5 ER subject to x1 + x2x5 x2 + x3 + x5 -2x2 + x4+x5 X1, X2, X3, X4, X5 = = 1 5 5 0 (3)
Example 12.14. Find an optimal solution to the following L.P.P. be computing all basic solutions and then finding one that maximizes the objective function : == 2x₂ + 3x₂-x3 + 4x4 = 8,x₁ - 2x₂ + 6x3-7x4 Max. Z = 2x₂ + 3x₂ + 4x3 + 7x4 Xp X X3 X4 ≥ 0, 9
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Consider the following LPP: Min Z = 3X1 + X2 + X3 Such that: X1 – 2X2 + X3 3 which has the following optimum solution: Basic| X, Xz Х Ха Хъ R R2 Solution Z-5 -1 1- M -1-M X4 3 1 -2 -5 12 X2 1 -1 -2 1 X3-2 10 0 1 Then, the dual optimal solution (yı, Y2, Y3, w) %3D Select one: (0,1,-1,2) (-1,1,-1,2) (0,0,1,1) (0,1,0,2)

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