Linear programming problem (simplex algorithm)

[SillyBen]

Can anyone help me to solve that problem ? I would really appreciate

For the linear programming problem P₁ :

Max z = 2 + x₁ - 2x₄
x₂ = 1 - x₄
x₃ = 0 - x₁ - x₄
x_i \(\displaystyle \ge\) 0
1 \(\displaystyle \le\) i \(\displaystyle \le\) 4

Question 1 :
Explain why the writing of that linear programming in the form proposed above allows one iteration of the simplex algorithm

Question 2 :
Give the associated solution with P₁ form above. You will yield the value of each variable and the objective function.

Question 3 :
Apply the simplex algorithm. You will give the value for each variable for the optimal solution after iteration.

Question 4 :
Compare results between question 2 and 3 solutions

Question 5 :
Infer from the previous question that the stopping criterion of the simplex algorithm is sufficient but not necessary condition.

 

[Valkarie]

Answer 1: The writing of the linear programming in the form proposed above allows one iteration of the simplex algorithm because it is expressed in the standard form. This means that all variables are greater than or equal to zero and that the objective function is a linear combination of the variables. The simplex algorithm can be applied to this form of linear programming, which is why it allows one iteration of the algorithm.Answer 2:The associated solution with P1 form above is x1 = 0, x2 = 1, x3 = 0, x4 = 1, and the objective function value is z = 2.Answer 3:Applying the simplex algorithm, the optimal solution after iteration is x1 = 0, x2 = 0, x3 = 1, x4 = 1 and the objective function value is z = 2.Answer 4:Comparing results between question 2 and 3 solutions, we see that the values for the variables are the same (x1 = 0, x2 = 0, x3 = 1, x4 = 1) but the objective function value is different (question 2: z = 2, question 3: z = 2). This is because the simplex algorithm optimizes the objective function value.Answer 5:Inferring from the previous question, we can conclude that the stopping criterion of the simplex algorithm is a sufficient but not necessary condition. This means that while the simplex algorithm will always find an optimal solution, it may not be the global optimal solution.