Question

The starting simplex table of a linear programming problem is given below, where \( S_1, S_2, S_3, S_4 \) are the slack variables. The objective of the problem is
Maximize \( z = 6x_1 + 4x_2 \)
The leaving variable among the basic variables is:

• A. \( S_1 \)
• B. \( S_2 \)
• C. \( S_3 \)
• D. \( S_4 \)

Hint:

In the simplex method, the leaving variable is found by comparing the ratios of the solution values to the pivot column's positive entries. The smallest ratio determines the leaving variable.

Answer 1

The Correct Option is A

In the simplex method, the objective is to identify the leaving variable in the basic feasible solution. The leaving variable is determined by the minimum ratio test. This test helps identify which basic variable will be replaced by the non-basic variable in the next iteration. We need to check the minimum positive ratio of the solution values to the corresponding coefficients in the \( x_1 \) column. 
The ratios are calculated as follows: 
- For \( S_1 \): \( \frac{36}{6} = 6 \) 
- For \( S_2 \): \( \frac{40}{2} = 20 \) 
- For \( S_3 \): \( \frac{2}{-1} \) (Negative value, not considered) 
- For \( S_4 \): \( \frac{3}{0} \) (Not valid, as division by zero is undefined) 
From these calculations, the minimum positive ratio is 6, which corresponds to \( S_1 \). Therefore, the leaving variable is \( S_1 \), which is option (A).