To solve a minimization problem in linear programming, the following key steps are involved:

1. Formulate the problem in standard form. This means that the objective function must be minimized and all constraints must be in the form of less-than-or-equal-to inequalities. If the problem is not already in standard form, it can be converted using the following steps:

   • Multiply both sides of the objective function by -1 if it is a maximization problem.
   • Convert any greater-than-or-equal-to inequalities to less-than-or-equal-to inequalities by subtracting a slack variable from each side of the inequality.
   • Add a non-negativity constraint for each variable.

2. Choose a solution method. There are a number of different solution methods available for linear programming problems, including the simplex method, the interior-point method, and the dual method. The most common method is the simplex method.

3. Solve the problem using the chosen solution method. The simplex method is an iterative algorithm that starts with a feasible solution and then iteratively moves to better feasible solutions until the optimal solution is reached.

4. Interpret the results. Once the problem has been solved, it is important to interpret the results correctly. The optimal solution will give the values of the decision variables that minimize the objective function. It is also important to check the feasibility of the optimal solution and to make sure that it satisfies all of the constraints.

Example

Consider the following minimization problem in linear programming:

Minimize: z = 2x + 3y
Subject to:
    x + y ≤ 5
    x ≥ 1
    y ≥ 1
    x ≥ 0
    y ≥ 0

To solve this problem using the simplex method, we first need to convert it to standard form. This gives us the following problem:

Minimize: z = -2x - 3y
Subject to:
    x + y + s1 = 5
    x - s2 = 1
    y - s3 = 1
    x ≥ 0
    y ≥ 0
    s1 ≥ 0
    s2 ≥ 0
    s3 ≥ 0

Next, we need to create a tableau. The tableau is a table that represents the current state of the problem. The initial tableau is shown below:

Basic | z | x | y | s1 | s2 | s3 | RHS
------- | -- | -- | -- | -- | -- | -- | --
s1      | 1 | 0 | 0 | 1 | 0 | 0 | 5
s2      | 0 | 1 | 0 | 0 | -1 | 0 | 1
s3      | 0 | 0 | 1 | 0 | 0 | -1 | 1
z       | -2 | -2 | -3 | 0 | 0 | 0 | 0

The first three rows of the tableau represent the constraints, and the last row represents the objective function. The RHS column shows the right-hand sides of the constraints.

We now need to use the simplex method to solve the problem. The simplex method works by iteratively replacing basic variables with non-basic variables until the optimal solution is reached.

In the first iteration, we will replace the non-basic variable x with the basic variable s2. To do this, we divide the second row by 1 and then subtract multiples of the second row from the other rows to make x zero in all rows except for the second row. The resulting tableau is shown below:

Basic | z | x | y | s1 | s2 | s3 | RHS
------- | -- | -- | -- | -- | -- | -- | --
x       | 1 | 0 | 0 | 1 | -1 | 0 | 1
s1      | 1 | 0 | 1 | 1 | 0 | 0 | 4
s3      | 0 | 0 | 1 | 0 | 0 | -1 | -1
z       | -2 | 2 | -3 | 0 | 0 | 0 | -2

The optimal solution is now reached, because the objective function coefficient for all non-basic variables is positive. The optimal solution is x = 1, y = 3, and z = -2.

Conclusion

The steps involved in solving a minimization problem in linear programming are as follows:

1. Formulate the problem in standard form.
2. Choose a solution method.
3. Solve the problem using the chosen solution method.
4. Interpret the results.

The simplex method