Consider the following linear programming problem (P) in standard form:(P) min{c"x : Ax = b, x > 0},where A E Rmxn¸ b€ R™, and c e R".Let ī E R" be a vertex of the feasible region of (P).(4.2) Let d e R" be a feasible direction at such that d + 0. Prove that –d e R" is not a feasibledirection at . (Hint: Consider the effect of the hypothesis d+0 on the subvector dN, whereBC {1,..., n} and N C {1,.. , n} denote the corresponding index sets of the vertex ē.)

Given: The linear programming problem (P) is the standard form is as follows, PmincTx:Ax=b, x≥10…

Answered: Consider the following linear…

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. Let u(x,y) = min (3x, 5y), p_x = 2, p_y = 5, and I = 20. Solve the constrained utility maximization problem, find x*, y*, and u*. Show it on the graph.
Solve the following nonlinear programming problem min c"r subject to a"Qx = d where Q = Q" > 0 and d > 0.
3.16 Solve the standard form linear programs with the following initial simplex tableaus: (b) X2 X4 - 1 - 1 1 1 -1 1 1 - 1 2 -3 1 -2
2. Rewrite the following linear program in standard form. (1) s.t. min z = -5x1 - 6x2 - 7x3 -x1 - 5x2 3x3 ≥ 15 -5x1 - 6x2 + 10x3 ≤ 20 x1x2x3 = -5 x1 ≤ 0, x2 > 0, x3 unconstrained -
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4.6. Consider the linear program minimize z = 2x1 - 3x2 subject to 4x1+3x2 < 12 X1 – 2x2 < 2 X1, X2 0. Represent the point x (1, 1)' as a convex combination of extreme points plus, if applicable, a direction of unboundedness. Find three different representations.
C. Consider the linear program max Xx1+ Bx2 s.t. X1 – x2 0, x2 > 0 where a E R and BER are real parameters. Determine all values of a and B for which this linear program (a) has a unique optimal solution; (b) has more than one optimal solution; (c) is unbounded.
c) A student wishes to allocate her available study time of 60 hours per week between two subjects in such a way as to maximize her grade average. She formulates two functions governing the grades of each module which can be taken as: 9₁ (t₁) = 20 + 20√₁, with t₁ > 0 and 9₂ (t₂) = -80 + 3t2, with t₂ > 80/3 maximize the grade average subject to the time constraint t₁ + t₂ = 60. You do not need to show that you have obtained a maximum.
1. Write the following linear program into standard form, then write the linear program corresponding to the first phase simplex: min x1,X2 ,X3ER 2x1 + x2 + X3 X1 – x2 + 2x3 < (1) subject to x1 + x2 – 3 -5 X1, X2, X3 T 2. Determine whether x = (0 2 0 1 1)ʻ is a vertex of max x1 + 2x2 – X4 + x5 X1,X2,X3,x4,X5ER xi + x2 -x2 + 3x3 + x4 -x2 + x3 – x4 + 4x5 (2) -1 subject to 1 X1; X2; X3, X4, X5 3. Determine whether x= (0 3 0 9 2)' is a solution to max 3x2 + 2x5 x1,x2,x3,x4,X5ER X1 + x2 – x5 X2 + x3 + x5 -2x2 + x4 + x5 (3) 5 subject to 5 X1, X2, X3, X4, X5 || || || A VI A
answer a to c
Consider the simplex tableau that is the final one in a problem to maximize x + 2y + 3z. xyzu v M 210-380 21 3015205 100651 19 What are the values of x, y and z when the maximum value of x + 2y + 3z occurs? A) x = 2, y = 5, z = 0. B) x = 19, y = 2, z = 5. C) x = 2, y = 1, z = 0. D) x = 0, y = 2, z = 5. E) none of these

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