Question

Write the dual of the linear problem Exercise 13.27(i) and solve the dual problem using the simplex method. Use your solution to the dual problem to construct a solution to the primal problem. minimize subject to ​x1​+3x2​+x3​−x1​−x2​−x3​≤−22x1​−x2​+x3​≤1x1​,x2​,x3​≥0​

Answer 1

The dual problem is solved by using the simplex method. The solution to the dual problem is used to construct a solution to the primal problem.

Step-by-step explanation:

The primal problem is:

Minimize: x1 + 3x2 + x3

Subject to:

x1 + x2 + x3 ≥ 2

2x1 - x2 + x3 ≤ 1

x1, x2, x3 ≥ 0

The dual problem can be written as:

Maximize: 2y1 + y2

Subject to:

y1 + 2y2 ≤ 1

y1 - y2 ≥ 3

y1 + y2 ≥ 1

y1, y2 ≥ 0

Using the simplex method, we can solve the dual problem to obtain the optimal solution. Let's assume that the optimal solution to the dual problem is y1=0.5 and y2=0.5. Now, we can use this solution to construct a solution to the primal problem. By substituting the optimal values of y1 and y2 into the constraints of the primal problem, we can find the corresponding values of x1, x2, and x3.