QuestionQuestion

1) Construct the canonical forms of the following linear programs.
(a) max { 3x + 2y } subject to:
2x - y ≤ 6;
2x + y ≤ 10;
x, y ≥ 0.
(b) max { x + y } subject to:
-2x + y ≤ 0;
x - 2y ≤ 0,
x + y ≤ 9;
x, y ≥ 0.
(c) min { 2x₁ - x₂ + 3x₃ } subject to:
x₁ - x₂ + 4x₃ ≤ 8;
2x₁ + 2x₂ - 5x₃ = 4;
x₁       + x₃ ≥ 6; x₁, x₃ ≥ 0, x₂ ≤ 0.
(d) max { x₁ + 2x₂ } subject to:
4x₁ - 2x₂ + 3x₃ ≤ 13;
5x₁ - 2x₂       ≥ 10;
x₁,x₂ ≥ 0.

2) Consider the system of equations Ax = B where
       [ 2 3 1 0 0]             [1]
A = [-1 1 0 2 1] and B = [1]
       [ 0 6 1 0 3]             [4]
Determine which of the following are basic solutions to the system.
(a) (1, 0,-1, 1, 0). (b) (0, 2,-5, 0, 1). (c) (0, 0, 1, 0, 1).

3) Suppose the canonical form of a linear program is given by the constraint matrix A and right-hand-side vector B:
       [3 0 1 1 0]             [5]
A = [2 1 0 0 0] and B = [3]
       [4 0 3 0 1]             [6]
Determine (and justify) which of the following solutions is
(i) a feasible solution to the linear programming problem.
(ii) an extreme point of the feasible region.
(iii) a basic solution.
(iv) a basic feasible solution.
For each basic feasible solution, list the basic variables.
(a) (0, 3, 0, 5, 6).
(b) (0, 3, 5, 0,-9).
(c) (3/2, 0, 0, 1/2, 0).
(d) (1/2, 1, 1, 0, 2).
(e) (1, 1, 1/2, 3/2, 1/2).

4) Consider the linear program
max { c'x + c"y }, subject to: x - y ≤ 2; 2x - y ≥ -4; x, y ≥ 0.
(a) Identify every extreme point of the feasible region.
(b) Identify every extreme direction of the feasible region.
(c) Characterize the set of values (c', c") for which there is a finite optimal solution.
(d) Characterize the set of values (c', c") for which there is an unbounded solution.

Solution PreviewSolution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

1) Explanation: canonical form needs equations with non-negative variables and initial basis, so every inequality should be equalized, every arbitrary variable should be expressed as a difference of other 2 non-negative variables (x = x’ – x”). Eventually artificial basic variables should be introduced, if simplex-method with artificial basis(M-method) will be used.
(a) max {3x + 2y + 0u + 0v}
          2x – y + u      = 6;
          2x + y +       v = 10;...

By purchasing this solution you'll be able to access the following files:
Solution.docx.

$10.50
for this solution

or FREE if you
register a new account!

PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

Find A Tutor

View available Operations Research Tutors

Get College Homework Help.

Are you sure you don't want to upload any files?

Fast tutor response requires as much info as possible.

Decision:
Upload a file
Continue without uploading

SUBMIT YOUR HOMEWORK
We couldn't find that subject.
Please select the best match from the list below.

We'll send you an email right away. If it's not in your inbox, check your spam folder.

  • 1
  • 2
  • 3
Live Chats