Question

Given a LP model as:Minimize Z = 2X1+ 4X2+ 6X3

Subject to:

X1+2X2+ X3≥2

X1–X3≥1

X2+X3= 1

2X1+ X2≤3

X2, X3 ≥0, X1 urs

a) Find the standard form of the LP problem.

b) Find the starting tableau to solve the Primal LP problem by using the M-Technique.

Answer #1

Consider the following
LP: Max Z=X1+5X2+3X3
s.t. X1+2X2+X3=3
2X1-X2 =4 X1,X2,X3≥0
a.) Write the associated dual model
b.) Given the information that the optimal basic variables are
X1 and X3, determine the associated optimal dual solution.

Consider the following problem.
Maximize Z = 2x1 -
x2 + x3,
subject to
x1 -
x2 + 3x3 ≤ 4
2x1 +
x2
≤ 10
x1 - x2 -
x3 ≤ 7
and x1 ≥
0, x2 ≥ 0,
x3 ≥ 0.
Use Excel Solver to solve this problem.
Write out the augmented form of this problem by introducing
slack variables.
Work through the simplex method step by step in tabular form to
solve the problem.

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)

Solve The LP problem using the graphic method
Z Max=5X1+3X2
Constaint function:
2X1 + 4X2 ≤ 80
5X1 + 2X2 ≤ 80
X1≥ 0 , X2≥0

Find the dual of the following LP, using direct method.
minz=4X1 +2X2 -X3
subject to
X1 +2X2 ≤6
X1 -X2 +2X3 =8
X1 ≥0,X2 ≥0,X3 urs

Solve the following LP model using the dual simplex method. Use
the format of the tabular form of the simplex without converting
the problem into a maximization problem.
Minimize -2x1 – x2
Subject to
x1+ x2+ x3 = 2
x1 + x4 = 1
x1, x2, x3, x4 ³ 0

Duality Theory: Consider the following LP:
max 2x1+2x2+4x3
x1−2x2+2x3≤−1
3x1−2x2+4x3≤−3
x1,x2,x3≤0
Formulate a dual of this linear program. Select all the correct
objective function and constraints
1. min −y1−3y2
2. min −y1−3y2
3. y1+3y2≤2
4. −2y1−2y2≤2
5. 2y1+4y2≤4
6. y1,y2≤0

Solve the LPP below by making use of the dual simplex
method.
min z=2x1+3x2+4x3
st: x1+2x2+x3>=3
2x1-x2+3x3>=4
x1,x2,x3>=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

2X1-X2+X3+7X4=0
-1X1-2X2-3X3-11X4=0
-1X1+4X2+3X3+7X4=0
a. Find the reduced row - echelon form of the coefficient
matrix
b. State the solutions for variables X1,X2,X3,X4 (including
parameters s and t)
c. Find two solution vectors u and v such that the solution
space is \
a set of all linear combinations of the form su + tv.

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