Solve the dual of a linear programming problem.

To solve the dual of a linear programming problem, we need to follow a few steps.

Firstly, we need to write the primal problem in standard form. This means that all variables must be non-negative and all constraints must be in the form of ≤.

Next, we need to write the dual problem. The dual problem has one variable for each constraint in the primal problem, and one constraint for each variable in the primal problem. The objective function of the dual problem is to maximise the sum of the products of the dual variables and the right-hand side values of the primal constraints.

After writing the dual problem, we need to convert it into standard form. This means that all variables must be non-negative and all constraints must be in the form of ≤.

Finally, we can solve the dual problem using any method we choose, such as the simplex method. The optimal solution to the dual problem gives us the optimal value of the primal problem.

It is important to note that the optimal solutions to the primal and dual problems are always equal. This is known as the strong duality theorem.

In summary, to solve the dual of a linear programming problem, we need to write the primal problem in standard form, write the dual problem, convert it into standard form, and solve it using any method we choose. The optimal solution to the dual problem gives us the optimal value of the primal problem.