Numerical Analysis Questions Long
The simplex method is a widely used algorithm for solving linear programming problems. It is an iterative process that starts with an initial feasible solution and systematically improves it until an optimal solution is found. The method operates on a polytope, which is a bounded region defined by a set of linear inequalities.
Here is a step-by-step description of the simplex method:
1. Formulate the linear programming problem in standard form, which involves maximizing or minimizing a linear objective function subject to a set of linear constraints.
2. Convert the problem into an augmented matrix form, known as the simplex tableau. The tableau consists of the coefficients of the variables, the right-hand side values of the constraints, and the objective function coefficients.
3. Identify the pivot column, which is the column with the most negative coefficient in the bottom row of the tableau. If all coefficients are non-negative, the current solution is optimal.
4. Determine the pivot row by selecting the row with the smallest non-negative ratio of the right-hand side value to the corresponding coefficient in the pivot column. This row will be the one that enters the basis.
5. Perform row operations to make the pivot element equal to 1 and all other elements in the pivot column equal to 0. This is achieved by dividing the pivot row by the pivot element and subtracting multiples of the pivot row from other rows.
6. Update the tableau by applying the row operations. The pivot row becomes the new basis row, and the pivot column becomes the new basis column.
7. Repeat steps 3 to 6 until an optimal solution is reached. This occurs when all coefficients in the bottom row of the tableau are non-negative.
8. Extract the optimal solution from the tableau. The values of the variables in the basis columns correspond to the optimal solution.
Now, let's consider an example to illustrate the simplex method:
Maximize: Z = 3x + 4y
Subject to:
2x + y ≤ 8
x + 2y ≤ 6
x, y ≥ 0
We first convert the problem into standard form by introducing slack variables:
Maximize: Z = 3x + 4y
Subject to:
2x + y + s1 = 8
x + 2y + s2 = 6
x, y, s1, s2 ≥ 0
The initial tableau is as follows:
| 2 1 1 0 8 |
| 1 2 0 1 6 |
| 3 4 0 0 0 |
The pivot column is the second column since it has the most negative coefficient in the bottom row. The pivot row is the second row since the ratio of the right-hand side value to the coefficient in the pivot column is smallest.
After performing row operations, the updated tableau is:
| 1 0 1/2 -1/2 4 |
| 0 1 -1/2 1/2 1 |
| 0 0 -3/2 3/2 -6 |
Since all coefficients in the bottom row are non-negative, the current solution is optimal. The optimal solution is x = 4, y = 1, with Z = 3(4) + 4(1) = 16.
In this example, the simplex method was used to find the optimal solution to the linear programming problem by iteratively improving the feasible solution until the objective function was maximized.