Question

convert the following problem into matrix format. use xij as slack variables for constrains

Max Z= x1+ x2+ 1.2x3+ 1.2x4+ 0.8x5

subjected to

2x1+ 2x3+ x4+ x5≤ 12

2x2+ x3+ 2x4+ x5≤ 15

x1+ x2+ x5≤ 15

5x1+ 7x2+ 4x3+ 5x4+ 6x5≤ 60

x1+ x2+ x3+ x4+ x5≤ 10

x1, x2, x3, x4, x5≥ 0

Answer #1

Find the fundamental system of solutions to the system.
2x1 − x2 + 3x3 + 2x4
+ x5 = 0
x1 + 4x2 − x4 + 3x5
= 0
2x1 + 6x2 − x3 + 5x4
= 0
5x1 + 9x2 + 2x3 +
6x4 + 4x5 = 0.

3. Consider the system of linear equations
3x1 + x2 + 4x3 − x4
= 7
2x1 − 2x2 − x3 + 2x4
= 1
5x1 + 7x2 + 14x3 −
8x4 = 20
x1 + 3x2 + 2x3 + 4x4
= −4
b) Solve this linear system applying Gaussian forward
elimination with partial pivoting and back ward substitution, by
hand. In (b) use fractions throughout your calculations.
(i think x1 = 1, x2= -1, x3 =1,
x4=-1, but i...

Solve for all 4-tuples (x1, x2, x3, x4) simultaneously
satisfying the following equations:
8x1 −9x2 −2x3 −5x4 = 100
9x1 +6x2 −6x3 +9x4 = 60
−3x1 −9x2 +4x3 −2x4 = −52
−7x2 +8x3 +8x4 = −135

what is the dual problem?
MAX 100X1+120X2+150X3+125X4
S.T.
1) X1 + 2X2 + 2X3 + 2X4 ≤ 108
2) 3X1 + 5X2 + X4 ≤ 120
3) X1 + X3 ≤ 25
4) X2 + X3 + X4 ≥ 50
X1,X2,X3,X4≥0

Find the duals of the following LP:
max z = 4x1 - x2 + 2x3
s.t. x1 + x2 <= 5
2x1 + x2 <= 7
2x2 + x3 >= 6
x1 + x3 = 4
x1 >=0, x2, x3 urs
show steps

Max Z = 2x1 + 8x2 + 4x3
subject to
2x1 + 3x2 ≤ 8
2x2 + 5x3 ≤ 12
3x1 + x2 + 4x3 ≤15
and x1,x2,x3≥0;
Verify that your primal and dual solutions are indeed optimal
using the Complementary Slackness theorem.

max Z = 5x1+3x2+x3
s.t : 2x1+x2+x3 < 6
x1+2x2+x3 < 7
x1, x2, x3 > 0
Solve the problem. What is the optimal value of the objective
function (OF)? Decision variables?
Solve the problem. What is the optimal value of the objective
function (OF)? Decision variables?
(20 points)

How do i generate a gomory cut from this optimal dictionary
form, expressing all the cuts in terms of the variables x1 and
x2:
max 15 - 4x3 - 2x4
s.t. x1 = 11/4 - (5/2)x3 + (1/4)x4
x2 = 2 + 2x3
x1, x2, x3, x4 >= 0

Find the standard matrix for the following transformation T : R
4 → R 3 : T(x1, x2, x3, x4) = (x1 − x2 + x3 − 3x4, x1 − x2 + 2x3 +
4x4, 2x1 − 2x2 + x3 + 5x4) (a) Compute T(~e1), T(~e2), T(~e3), and
T(~e4). (b) Find an equation in vector form for the set of vectors
~x ∈ R 4 such that T(~x) = (−1, −4, 1). (c) What is the range of
T?

Solve the linear programs using the simplex tableau.
Max
Z = -6X1 - 14X2 - 13X3
Subject to X1 +
4X2 + 2X3 ≤ 48
X1 + 2X2 + 4X3 ≤ 60
X1, X2, X3 ≥ 0

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